Recent Trends in Special Numbers and Special Functions and

International Journal of Mathematics and Mathematical Sciences

Guest Editors Serkan Araci Mehmet Acikgoz Cenap Oumlzel H M Srivastava and Toka Diagana

Recent Trends in Special Numbers and Special Functions and Polynomials

Recent Trends in Special Numbers andSpecial Functions and Polynomials

International Journal of Mathematics andMathematical Sci-ences


Guest Editors Serkan Araci Mehmet Acikgoz Cenap OzelH M Srivastava and Toka Diagana

Copyright copy 2015 Hindawi Publishing Corporation All rights reserved

This is a special issue published in ldquoInternational Journal of Mathematics andMathematical Sciencesrdquo All articles are open access articlesdistributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work is properly cited

Editorial BoardThe editorial board of the journal is organized into sections that correspond tothe subject areas covered by the journal

Algebra

Adolfo Ballester-Bolinches SpainPeter Basarab-Horwath SwedenHoward E Bell CanadaMartin Bohner USATomasz Brzezinski UKRaul E Curto USADavid E Dobbs USA

Dalibor Froncek USAHeinz Peter Gumm GermanyPentti Haukkanen FinlandPalle E Jorgensen USAV R Khalilov RussiaAloys Krieg GermanyRobert H Redfield USA

Frank C Sommen BelgiumYucai Su ChinaChun-Lei Tang ChinaRam U Verma USADorothy I Wallace USASiamak Yassemi IranKaiming Zhao Canada

Geometry

Alberto Cavicchioli ItalyChristian Corda ItalyJerzy Dydak USABrigitte Forster-Heinlein GermanyS M Gusein-Zade Russia

Henryk Hudzik PolandR Lowen BelgiumFrdric Mynard USAHernando Quevedo MexicoMisha Rudnev UK

Nageswari Shanmugalingam USAZhongmin Shen USANistor Victor USALuc Vrancken France

Logic and Set Theory

Razvan Diaconescu RomaniaMessoud Efendiev GermanySiegfried Gottwald Germany

Balasubramaniam Jayaram IndiaRadko Mesiar SlovakiaSusana Montes-Rodriguez Spain

Andrzej Skowron PolandSergejs Solovjovs Czech RepublicRichard Wilson Mexico

Mathematical Analysis

Peter W Bates USAHeinrich Begehr GermanyOscar Blasco SpainTeodor Bulboaca RomaniaWolfgang zu Castell GermanyShih-sen Chang ChinaCharles E Chidume NigeriaHi Jun Choe Republic of KoreaRodica D Costin USAPrabir Daripa USAH S V De Snoo The NetherlandsLokenath Debnath USASever Dragomir AustraliaRicardo Estrada USAXianguo Geng ChinaAttila Gilanyi HungaryJerome A Goldstein USA

Narendra K Govil USAWeimin Han USASeppo Hassi FinlandHelge Holden NorwayNawab Hussain Saudi ArabiaPetru Jebelean RomaniaShyam L Kalla IndiaIrena Lasiecka USAYuri Latushkin USABao Qin Li USASongxiao Li ChinaNoel G Lloyd UKRaul F Manasevich ChileBiren N Mandal IndiaRam N Mohapatra USAManfred Moller South AfricaEnrico Obrecht Italy

Gelu Popescu USAJean Michel Rakotoson FranceB E Rhoades USAPaolo Emilio Ricci ItalyNaseer Shahzad Saudi ArabiaMarianna A Shubov USAHarvinder S Sidhu AustraliaLinda R Sons USAIlya M Spitkovsky USAMarco Squassina ItalyHari M Srivastava CanadaPeter Takac GermanyMichael M Tom USAIngo Witt GermanyA Zayed USAYuxi Zheng USA

Operations Research

Shih-Pin Chen TaiwanTamer Eren TurkeyO Hernandez-Lerma MexicoImed Kacem France

Yan K Liu ChinaMihai Putinar USAShey-Huei Sheu TaiwanTheodore E Simos Greece

Frank Werner GermanyChin-Chia Wu Taiwan

Probability and Statistics

Kenneth S Berenhaut USAJewgeni Dshalalow USAHans Engler USA

Serkan Eryılmaz TurkeyVladimir V Mityushev PolandAndrew Rosalsky USA

Gideon Schechtman IsraelNiansheng Tang ChinaAndrei I Volodin Canada

Contents

Recent Trends in Special Numbers and Special Functions and Polynomials Serkan AraciMehmet Acikgoz Cenap Ozel H M Srivastava and Toka DiaganaVolume 2015 Article ID 573893 1 page

The Peak of Noncentral Stirling Numbers of the First Kind Roberto B Corcino Cristina B Corcinoand Peter John B AranasVolume 2015 Article ID 982812 7 pages

ADomain-Specific Architecture for Elementary Function Evaluation Anuroop Sharmaand Christopher Kumar AnandVolume 2015 Article ID 843851 8 pages

Explicit Formulas for Meixner Polynomials Dmitry V Kruchinin and Yuriy V ShablyaVolume 2015 Article ID 620569 5 pages

Faber Polynomial Coefficients of Classes of Meromorphic Bistarlike Functions Jay M Jahangiriand Samaneh G HamidiVolume 2015 Article ID 161723 5 pages

Some Properties of Certain Class of Analytic Functions Uzoamaka A Ezeafulukwe and Maslina DarusVolume 2014 Article ID 358467 5 pages

A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized HypergeometricFunctions F Ghanim and M DarusVolume 2014 Article ID 374821 6 pages

New Expansion Formulas for a Family of the 120582-Generalized Hurwitz-Lerch Zeta FunctionsH M Srivastava and Sebastien GabouryVolume 2014 Article ID 131067 13 pages

EditorialRecent Trends in Special Numbers and SpecialFunctions and Polynomials

Serkan Araci1 Mehmet Acikgoz2 Cenap Oumlzel3 H M Srivastava4 and Toka Diagana5

1Department of Economics Faculty of Economics Administrative and Social Sciences Hasan Kalyoncu University27410 Gaziantep Turkey2Department of Mathematics Faculty of Arts and Sciences University of Gaziantep 27310 Gaziantep Turkey3Department of Mathematics Dokuz Eylul University 35160 Izmir Turkey4Department of Mathematics and Statistics University of Victoria Victoria BC Canada V8W 3R45Department of Mathematics Howard University 2441 6th Street NW Washington DC 20059 USA

Correspondence should be addressed to Serkan Araci mtsrknhotmailcom

Received 1 October 2015 Accepted 1 October 2015

Copyright copy 2015 Serkan Araci et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Special numbers and polynomials play an extremely impor-tant role in the development of several branches ofmathemat-ics physics and engineeringThey havemany algebraic oper-ations Because of their finite evaluation schemes and closureunder addition multiplication differentiation integrationand composition they are richly utilized in computationalmodels of scientific and engineering problems This issuecontributes to the field of special functions and polynomialsAn importance is placed on vital and important develop-ments in classical analysis number theory mathematicalanalysis mathematical physics differential equations andother parts of the natural sciences

One of the aims of this special issue was to surveyspecial numbers special functions and polynomials wherethe essentiality of the certain class of analytic functionsgeneralized hypergeometric functions Hurwitz-Lerch Zetafunctions Faber polynomial coefficients the peak of noncen-tral Stirling numbers of the first kind and structure betweenengineering mathematics are highlighted

All manuscripts submitted to this special issue are sub-jected to a quick and closed peer-review process The guesteditors initially examined themanuscripts to check suitabilityof papers Based on referees who are well-known mathe-maticians of this special issue we got the best articles to beincluded in this issue The results and properties of acceptedpapers are very interesting well defined and mathematically

correct The work is a relevant contribution to the fields ofanalysis and number theory

Serkan AraciMehmet Acikgoz

Cenap OzelH M Srivastava

Toka Diagana

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2015 Article ID 573893 1 pagehttpdxdoiorg1011552015573893

Research ArticleThe Peak of Noncentral Stirling Numbers of the First Kind

Roberto B Corcino1 Cristina B Corcino1 and Peter John B Aranas2

1Mathematics and ICT Department Cebu Normal University 6000 Cebu City Philippines2Department of Mathematics Mindanao State University Main Campus 9700 Marawi City Philippines

Correspondence should be addressed to Roberto B Corcino rcorcinoyahoocom

Received 18 September 2014 Accepted 20 November 2014

Academic Editor Serkan Araci

Copyright copy 2015 Roberto B Corcino et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We locate the peak of the distribution of noncentral Stirling numbers of the first kind by determining the value of the indexcorresponding to the maximum value of the distribution

1 Introduction

In 1982 Koutras [1] introduced the noncentral Stirling num-bers of the first and second kind as a natural extensionof the definition of the classical Stirling numbers namelythe expression of the factorial (119909)

119899in terms of powers of

119909 and vice versa These numbers are respectively denotedby 119904

119886(119899 119896) and 119878

119886(119899 119896) which are defined by means of the

following inverse relations

(119905)119899=

119899

sum119896=0

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

(119905 minus 119886)119896

(1)

(119905 minus 119886)119899

=

119899

sum119896=0

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(119905)119896 (2)

where 119886 119905 are any real numbers 119899 is a nonnegative integerand

119904119886(119899 119896) =

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

119878119886(119899 119896) =

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(3)

The numbers satisfy the following recurrence relations

119904119886(119899 + 1 119896) = 119904

119886(119899 119896 minus 1) + (119886 minus 119899) 119904

119886(119899 119896) (4)

119878119886(119899 + 1 119896) = 119878

119886(119899 119896 minus 1) + (119896 minus 119886) 119878

119886(119899 119896) (5)

and have initial conditions

119904119886(0 0) = 1 119904

119886(119899 0) = (119886)

119899 119904

119886(0 119896) = 0 119899 119896 = 0

119878119886(0 0) = 1 119878

119886(119899 0) = (minus119886)

119899

119878119886(0 119896) = 0 119899 119896 = 0

(6)

It is worth mentioning that for a given negative binomialdistribution 119884 and the sum 119883 = 119883

1+ 119883

2+ sdot sdot sdot + 119883

119896of

119896 independent random variables following the logarithmicdistribution the numbers 119904

119886(119899 119896) appeared in the distribu-

tion of the sum 119882 = 119883 + 119884 while the numbers 119878119886(119899 119896)

appeared in the distribution of the sum 119885 = 119883 + where119883 is the sum of 119896 independent random variables followingthe truncated Poisson distribution away from zero and isa Poisson random variable More precisely the probabilitydistributions of119882 and 119885 are given respectively by

119875 [119882 = 119899] =119896

(1 minus 120579)minus119904

(minus log (1 minus 120579))119896

120579119899

119899(minus1)

119899minus119896

119904minus119904(119899 119896)

119875 [119885 = 119899] =119896

119890119898 (119890119897 minus 1)119896

119897119899

119899(minus1)

119899minus119896

119878minus119898119897

(119899 119896)

(7)

For a more detailed discussion of noncentral Stirling num-bers one may see [1]

Determining the location of the maximum of Stirlingnumbers is an interesting problem to consider In [2] Mezo

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2015 Article ID 982812 7 pageshttpdxdoiorg1011552015982812

2 International Journal of Mathematics and Mathematical Sciences

obtained results for the so-called 119903-Stirling numbers whichare natural generalizations of Stirling numbers He showedthat the sequences of 119903-Stirling numbers of the first andsecond kinds are strictly log-concave Using the theorem ofErdos and Stone [3] he was able to establish that the largestindex for which the sequence of 119903-Stirling numbers of the firstkind assumes its maximum is given by the approximation

119870(1)

119899119903= 119903 + [log(119899 minus 1

119903 minus 1) minus

1

119903+ 119900 (1)] (8)

Following the methods of Mezo we establish strict log-concavity and hence unimodality of the sequence of noncen-tral Stirling numbers of the first kind and eventually obtainan estimating index at which the maximum element of thesequence of noncentral Stirling numbers of the first kindoccurs

2 Explicit Formula

In this section we establish an explicit formula in symmetricfunction form which is necessary in locating the maximumof noncentral Stirling numbers of the first kind

Let 119891119894(119909) 119894 = 1 2 119899 be differentiable functions and let

119865119899(119909) = prod

119899

119894=1119891119894(119909) It can easily be verified that for all 119899 ge 3

1198651015840

119899(119909) = sum

1le1198951lt1198952ltsdotsdotsdotlt119895

119899minus1le119899

119894isinN11989911989511198952119895119899minus1

1198911015840

119894(119909)

119899minus1

prod119896=1

119891119895119896(119909) (9)

Now consider the following derivative of (120585119895 + 119886)119899when 119899 =

1 2

119889

119889120585(120585119895 + 119886)

1= 119895

119889

119889120585(120585119895 + 119886)

2= (120585119895 + 119886) 119895 + (120585119895 + 119886 minus 1) 119895

(10)

Then for 119899 ge 3 and using (9) we get

1

119895119899119889

119889120585(120585119895 + 119886)

119899= sum0le1198951lt1198952ltsdotsdotsdotlt119895

119899minus1le119899minus1

119899minus1

prod119896=1

(120585 +119886 minus 119895

119896

119895) (11)

Then we have the following lemma

Lemma 1 For any nonnegative integers 119899 and 119896 one has

1

119895119899119889119896

119889120585119896(120585119895 + 119886)


119899minus119896le119899minus1

119896

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(12)

Proof We prove by induction on 119896 For 119896 = 0 (12) clearlyholds For 119896 = 1 (12) can easily be verified using (11) Supposefor119898 ge 1

1

119895119899119889119898

119889120585119898(120585119895 + 119886)


119899minus119898le119899minus1

119898

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(13)

Then

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899

= 119898 sum0le1198951lt1198952ltsdotsdotsdotlt119895

119899minus119898le119899minus1

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(14)

where the sum has ( 119899119899minus119898

) = 119899(119899 minus 1)(119899 minus 2) sdot sdot sdot (119899 minus119898 + 1)119898

terms and its summand

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895) = sum

1198941lt1198942ltsdotsdotsdotlt119894119899minus119898minus1

119899minus119898minus1

prod119902=1

(120585 +119886 minus 119894

119902

119895)

(15)

119894119902

isin 1198951 1198952 119895

119899minus119898 has ( 119899minus119898

119899minus119898minus1) = (119899 minus 119898)(119899 minus 119898 minus

1)(119899 minus 119898 minus 1) = 119899 minus 119898 terms Therefore the expansionof (119889119898+1119889120585119898+1)(120585119895 + 119886)

119899has a total of 119899(119899 minus 1) sdot sdot sdot (119899 minus

119898 + 1)(119899 minus 119898)119898 terms of the form prod119899minus119898minus1

119902=1(120585 + (119886 minus

119895119902)119895) However if the sum is evaluated over all possible

combinations 11989511198952sdot sdot sdot 119895

119899minus119898minus1such that 0 le 119895

1lt 1198952lt sdot sdot sdot lt

119895119899minus119898minus1

le 119899 minus 1 then the sum has ( 119899

119899minus119898minus1) = 119899(119899 minus 1) sdot sdot sdot (119899 minus

119898)(119899 minus 119898 minus 1)(119898 + 1)(119899 minus 119898 minus 1) = (1(119898 + 1))(119899(119899 minus

1) sdot sdot sdot (119899 minus 119898)119898) distinct terms It follows that every termprod119899minus119898minus1

119902=1(120585 + (119886 minus 119895

119902)119895) appears119898 + 1 times in the expansion

of (1119895119899)(119889119898+1119889120585119898+1)(120585119895 + 119886)119899 Thus we have

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899


119899minus119898minus1le119899minus1

119898 + 1

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

= (119898 + 1) sum0le1198951lt1198952ltsdotsdotsdotlt119895


119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(16)

Lemma 2 Let 119904(119899 119896 119886) =

(1119896)lim120585rarr0

((sum119896

119895=0(minus1)

119896minus119895

( 119896119895) (119889119896119889120585119896)(120585119895 + 119886)

119899)119896)

Then

119904 (119899 119896 119886) = sum0le1198951lt1198952ltsdotsdotsdotlt119895

119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (17)

Proof Using Lemma 1

119904 (119899 119896 119886)

=1

119896lim120585rarr0

((

119896

sum119895=0

(minus1)119896minus119895

(119896

119895) 119895119899

sum0le1198951ltsdotsdotsdotlt119895

119899minus119896le119899minus1

119896

times

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)) (119896)

minus1

)

(18)

International Journal of Mathematics and Mathematical Sciences 3

Note that prod119899minus119896119902=1

(120585 + (119886 minus 119895119902)119895) = (1119895119899minus119896)prod

119899minus119896

119902=1(120585119895 + 119886 minus 119895

119902)

Hence the expression at the right-hand side of (18) becomes

1

119896lim120585rarr0

[

[

119896

sum119895=0


(119896

119895) 119895119896

times sum0le1198951lt1198952ltsdotsdotsdotlt119895

119899minus119896le119899minus1

119899minus119896

prod119902=1

(120585119895 + 119886 minus 119895119902)]

]

=1

119896[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)]

]

(19)

which boils down to

sum0le1198951lt1198952ltsdotsdotsdotlt119895

119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (20)

since

1

119896

119896

sum119895=0


(119896

119895) 119895119896

= 119878 (119896 119896) = 1 (21)

where 119878(119899 119896) denote the Stirling numbers of the second kind

Theorem 3 The noncentral Stirling numbers of the first kindequal

119904119886(119899 119896) = 119904 (119899 119896 119886) = sum


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (22)

Proof We know that


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (23)

is equal to the sumof the products (119886minus1198951)(119886minus119895

2) sdot sdot sdot (119886minus119895

119899+1minus119896)

where the sum is evaluated overall possible combinations11989511198952sdot sdot sdot 119895

119899+1minus119896 119895119894isin 0 1 2 119899 These possible combina-

tions can be divided into two the combinations with 119895119894= 119899

for some 119894 isin 1 2 119899 minus 119896 + 1 and the combinations with119895119894

= 119899 for all 119894 isin 0 1 2 119899 minus 119896 + 1 Thus


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (24)

is equal to


119899minus119896+1le119899minus1

119899minus119896+1

prod119902=1

(119886 minus 119895119902) + (119886 minus 119899)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)

(25)

This implies that

119904 (119899 + 1 119896 119886) = 119904 (119899 119896 minus 1 119886) + (119886 minus 119899) 119904 (119899 119896 119886) (26)

This is exactly the triangular recurrence relation in (4) for119904119886(119899 119896) This proves the theorem

The explicit formula inTheorem 3 is necessary in locatingthe peak of the distribution of noncentral Stirling numbers ofthe first kind Besides this explicit formula can also be usedto give certain combinatorial interpretation of 119904

119886(119899 119896)

A 0-1 tableau as defined in [4] by deMedicis and Lerouxis a pair 120593 = (120582 119891) where

120582 = (1205821ge 120582

2ge sdot sdot sdot ge 120582

119896) (27)

is a partition of an integer119898 and119891 = (119891119894119895)1le119895le120582

119894

is a ldquofillingrdquo ofthe cells of corresponding Ferrers diagram of shape 120582with 0rsquosand 1rsquos such that there is exactly one 1 in each column Usingthe partition 120582 = (5 3 3 2 1) we can construct 60 distinct 0-1 tableaux One of these 0-1 tableaux is given in the followingfigure with 119891

14= 11989115

= 11989123

= 11989131

= 11989142

= 1 119891119894119895= 0 elsewhere

(1 le 119895 le 120582119894)

0 0 0 1 10 0 11 0 00 10

(28)

Also as defined in [4] an 119860-tableau is a list 120601 of column 119888

of a Ferrers diagram of a partition 120582 (by decreasing order oflength) such that the lengths |119888| are part of the sequence 119860 =

(119886119894)119894ge0

119886119894isin 119885+ cup 0 If 119879119889119860(ℎ 119903) is the set of119860-tableaux with

exactly 119903 distinct columns whose lengths are in the set 119860 =

1198860 1198861 119886

ℎ then |119879119889119860(ℎ 119903)| = ( ℎ+1

119903) Now transforming

each column 119888 of an 119860-tableau in 119879119889119860(119899 minus 1 119899 minus 119896) into acolumn of length 120596(|119888|) we obtain a new tableau which iscalled 119860

120596-tableau If 120596(|119888|) = |119888| then the 119860

120596-tableau is

simply the 119860-tableau Now we define an 119860120596(0 1)-tableau to

be a 0-1 tableau which is constructed by filling up the cells ofan 119860

120596-tableau with 0rsquos and 1rsquos such that there is only one 1 in

each column We use 119879119889119860120596(01)(119899 minus 1 119899 minus 119896) to denote the setof such 119860

120596(0 1)-tableaux

It can easily be seen that every (119899 minus 119896) combination11989511198952sdot sdot sdot 119895

119899minus119896of the set 0 1 2 119899 minus 1 can be represented

geometrically by an element 120601 in 119879119889119860(119899 minus 1 119899 minus 119896) with 119895119894

as the length of (119899 minus 119896 minus 119894 + 1)th column of 120601 where 119860 =

0 1 2 119899 minus 1 Hence with 120596(|119888|) = 119886 minus |119888| (22) may bewritten as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

prod119888isin120601

120596 (|119888|) (29)

Thus using (29) we can easily prove the following theorem

Theorem4 Thenumber of119860120596(0 1)-tableaux in119879119889119860120596(01)(119899minus

1 119899minus119896) where119860 = 0 1 2 119899 minus 1 such that 120596(|119888|) = 119886minus |119888|

is equal to 119904119886(119899 119896)


Let 120601 be an 119860-tableau in 119879119889119860(119899 minus 1 119899 minus 119896) with 119860 =

0 1 2 119899 minus 1 and

120596119860(120601) = prod

119888isin120601

120596 (|119888|)

=

119899minus119896

prod119894=1

(119886 minus 119895119894) 119895

119894isin 0 1 2 119899 minus 1

(30)

If 119886 = 1198861+ 1198862for some 119886

1and 119886

2 then with 120596lowast(119895) = 119886

2minus 119895

120596119860(120601) =

119899minus119896

prod119894=1

(1198861+ 120596lowast

(119895119894))

=

119899minus119896

sum119903=0

119886119899minus119896minus119903

1sum


119903le119895119899minus119896

119903

prod119894=1

120596lowast

(119902119894)

(31)

Suppose 119861120601is the set of all 119860-tableaux corresponding to 120601

such that for each 120595 isin 119861120601either

120595 has no column whose weight is 1198861 or

120595 has one column whose weight is 1198861 or

120595 has 119899 minus 119896 columns whose weights are 1198861 Then we

may write

120596119860(120601) = sum

120595isin119861120601

120596119860(120595) (32)

Now if 119903 columns in 120595 have weights other than 1198861 then

120596119860(120595) = 119886

119899minus119896minus119903

1

119903

prod119894=1

120596lowast

(119902119894) (33)

where 1199021 1199022 119902

119903isin 119895

1 1198952 119895

119899minus119896 Hence (29) may be

written as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

sum120595isin119861120601

120596119860(120595) (34)

Note that for each 119903 there correspond ( 119899minus119896119903) tableaux with 119903

distinct columns havingweights119908lowast(119902119894) 119902119894isin 119895

1 1198952 119895

119899minus119896

Since119879119889119860(119899minus1 119899minus119896) has ( 119899119896) elements for each 120601 isin 119879119889119860(119899minus

1 119899 minus 119896) the total number of 119860-tableaux 120595 corresponding to120601 is

(119899

119896)(

119899 minus 119896

119903) (35)

elements However only ( 119899119903) tableaux in 119861

120601with 119903 distinct

columns of weights other than 1198861are distinct Hence every

distinct tableau 120595 appears

( 119899119896) ( 119899minus119896

119903)

( 119899119903)

= (119899 minus 119903

119896) (36)

times in the collection Consequently we obtain

119904119886(119899 119896) =

119899minus119896

sum119903=0

(119899 minus 119903

119896) 119886119899minus119896minus119903

1sum120595isin119861119903


120596lowast

(|119888|) (37)

where 119861119903denotes the set of all tableaux 120595 having 119903 distinct

columns whose lengths are in the set 0 1 2 119899 minus 1Reindexing the double sum we get

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

1sum

120595lowastisin119861119899minus119895

prod119888isin120595lowast

120596lowast

(|119888|) (38)

Clearly 119861119899minus119895

= 119879119889119860(119899 minus 1 119899 minus 119895) Thus using (22) we obtainthe following theorem

Theorem 5 The numbers 119904119886(119899 119896) satisfy the following iden-

tity

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

11199041198862

(119899 119895) (39)

where 119886 = 1198861+ 1198862for some numbers 119886

1and 119886

2

The next theorem contains certain convolution-type for-mula for 119904

119886(119899 119896) which will be proved using the combina-

torics of 119860-tableau

Theorem 6 The numbers 119904119886(119899 119896) have convolution formula

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (40)

Proof Suppose that 1206011is a tableau with exactly119898minus119896 distinct

columns whose lengths are in the set1198601= 0 1 2 119898 minus 1

and1206012is a tableauwith exactly 119895minus119899+119896distinct columnswhose

lengths are in the set 1198602= 119898119898 + 1119898 + 2 119898 + 119895 minus 1

Then 1206011isin 1198791198891198601(119898minus1119898minus119896) and 120601

2isin 1198791198891198602(119895 minus 1 119895 minus 119899+ 119896)

Notice that by joining the columns of 1206011and 120601

2 we obtain an

119860-tableau 120601 with 119898 + 119895 minus 119899 distinct columns whose lengthsare in the set 119860 = 0 1 2 119898 + 119895 minus 1 that is 120601 isin 119879119889

119860

(119898 +

119895 minus 1119898 + 119895 minus 119899) Hence

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601)

=

119899

sum119896=0

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011)

times

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

(41)


Note that

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

= sum119898le1198921lt1198922ltsdotsdotsdotlt119892

119895minus119899+119896le119898+119895minus1

119895minus119899+119896

prod119902=1

(119886 minus 119892119902)



119895minus119899+119896

prod119902=1

(119886 minus (119898 + 119892119902))

= 119904119886minus119898

(119895 119899 minus 119896)

(42)

Also using (29) we have

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011) = 119904

119886(119898 119896)

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601) = 119904

119886(119898 + 119895 119899)

(43)

Thus

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (44)

The following theorem gives another form of convolutionformula

Theorem 7 The numbers 119904119886(119899 119896) satisfy the second form of

convolution formula

119904119886(119899 + 1119898 + 119895 + 1) =

119899

sum119896=0

119904119886(119896 119899) 119904

119886minus(119896+1)(119899 minus 119896 119895) (45)

Proof Let

1206011be a tableau with 119896minus119898 columns whose lengths are

in 1198601= 0 1 119896 minus 1

1206012be a tableau with 119899 minus 119896 minus 119895 columns whose lengths

are in 1198602= 119896 + 1 119899

Then 1206011isin 1198791198891198601(119896 minus 1 119896 minus 119898) 120601

2isin 1198791198891198602(119899 minus 119896 minus 1 119899 minus 119896 minus

119895) Using the same argument above we can easily obtain theconvolution formula

3 The Maximum of Noncentral StirlingNumbers of the First Kind

We are now ready to locate the maximum of 119904119886(119899 119896) First let

us consider the following theorem on Newtonrsquos inequality [5]which is a good tool in proving log-concavity or unimodalityof certain combinatorial sequences

Theorem 8 If the polynomial 1198861119909+119886

21199092 + sdot sdot sdot + 119886

119899119909119899 has only

real zeros then

1198862

119896ge 119886119896+1

119886119896minus1

119896

119896 minus 1

119899 minus 119896 + 1

119899 minus 119896(119896 = 2 119899 minus 1) (46)

Now consider the following polynomial119899

sum119896=0

119904119886(119899 119896) (119905 + 119886)

119896

(47)

This polynomial is just the expansion of the factorial⟨119905⟩119899

= 119905(119905 + 1)(119905 + 2) sdot sdot sdot (119905 + 119899 minus 1) which has real roots0 minus1 minus2 minus119899+1 If we replace 119905 by 119905minus119886 we see at once thatthe roots of the polynomialsum119899

119896=0119904119886(119899 119896)119905119896 are 119886 119886minus1 119886minus

119899 + 1 Applying Newtonrsquos Inequality completes the proof ofthe following theorem

Theorem 9 The sequence 119904119886(119899 119896)

119899

119896=0is strictly log-concave

and hence unimodal

By replacing 119905 with minus119905 the relation in (1) may be writtenas

⟨119905⟩119899=

119899

sum119896=0


119904119886(119899 119896) (119905 + 119886)

119896

(48)

where ⟨119905⟩119899= 119905(119905 + 1)(119905 + 2) sdot sdot sdot (119905 + 119899 minus 1) Note that from

Theorem 3 with 119886 lt 0

119904119886(119899 119896) = (minus1)

119899minus119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902) (49)

where 119887 = minus119886 gt 0 Now we define the signless noncentralStirling number of the first kind denoted by |119904

119886(119899 119896)| as

1003816100381610038161003816119904119886 (119899 119896)1003816100381610038161003816 = (minus1)

119899minus119896

119904119886(119899 119896)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902)

(50)

To introduce themain result of this paper we need to statefirst the following theorem of Erdos and Stone [3]

Theorem 10 (see [3]) Let 1199061

lt 1199062

lt sdot sdot sdot be an infinitesequence of positive real numbers such that

infin

sum119894=1

1

119906119894

= infin

infin

sum119894=1

1

1199062119894

lt infin (51)

Denote by sum119899119896

the sum of the product of the first 119899 of themtaken 119896 at a time and denote by 119870

119899the largest value of 119896 for

which sum119899119896

assumes its maximum value Then

119870119899= 119899 minus [

119899

sum119894=1

1

119906119894

minus

119899

sum119894=1

1

1199062119894

(1 +1

119906119894

)

minus1

+ 119900 (1)] (52)

We also need to recall the asymptotic expansion ofharmonic numbers which is given by

1

1+1

2+ sdot sdot sdot +

1

119899= log 119899 + 120574 + 119900 (1) (53)

where 120574 is the Euler-Mascheroni constantThe following theorem contains a formula that deter-

mines the value of the index corresponding to the maximumof the sequence |119904

119886(119899 119896)|

119899

119896=0


Theorem 11 The largest index for which the sequence|119904119886(119899 119896)|

119899

119896=0assumes its maximum is given by the approxi-

mation

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (54)

where [119909] is the integer part of 119909 and 119887 = minus119886 119886 lt 0

Proof Using Theorem 10 and by (50) we see that |119904119886(119899 119896)| =

sum119899minus1119899minus119896

Denoting by 119896119886119899

for which sum119899119899minus119896

is maximum andwith 119906

1= 119887 + 0 119906

2= 119887 + 1 119906

119899minus1= 119887 + 119899 minus 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894)2(1 +

1

119887 + 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894) (119887 + 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894 + 1+ 119900 (1)]

(55)

But using (53) we see that

119899minus1

sum119894=0

1

119887 + 119894 + 1= log (119887 + 119899) minus log 119887 (56)

From this we get

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (57)

For the case in which 119886 gt 0 we will only consider thesequence of noncentral Stirling numbers of the first kind forwhich 119886 ge 119899

Theorem 12 The maximizing index for which the maximumnoncentral Stirling number occurs for 119886 ge 119899 is given by theapproximation

119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (58)

Proof From the definition for 119886 ge 119899119904119886(119899 119896) gt 0 and by

Theorem 3 119904119886(119899 119896) is the sum of the products (119886 minus 119895

1)(119886 minus


119899minus119896)where 119895

119894rsquos are taken from the set 0 1 2 119899minus

1 By Theorem 10 119904119886(119899 119896) = sum

119899119899minus119896 Thus with 119906

1= 119886 119906

2=

119886 minus 1 119906119899minus1

= 119886 minus 119899 + 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1

119886 minus 119894minus

119899minus1

sum119894=0

1

(119886 minus 119894)2(1 +

1

119886 minus 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894) (119886 minus 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119886 minus 119894 + 1+ 119900 (1)]

(59)

Table 1 Values of 119904minus1(119899 119896)

119899119896 0 1 2 30 11 12 2 13 6 6 14 24 35 10 15 120 225 85 156 720 1624 735 1757 5040 13132 6769 19608 40320 109584 118124 672849 362880 1026576 1172700 72368010 3628800 10628640 12753576 8409500

Again using (53) we get

119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (60)

Example 13 The maximum element of the sequence119904minus1(9 119896)

9

119896=0occurs at (Table 1)

119896minus19

= [log(1 + 8

1) + 119900 (1)]

= [log 9 + 119900 (1)]

asymp 2

(61)

Example 14 The maximum element of the sequence11990410(10 119896)

10


1198961010

= [log( 10 + 1

10 minus 10 + 1) + 119900 (1)]

= [log 11 + 119900 (1)]

asymp 2

(62)

We know that the classical Stirling numbers of the firstkind are special cases of 119904

119886(119899 119896) by taking 119886 = 0 However

formulas in Theorems 11 and 12 do not hold when 119886 = 0Hence these formulas are not applicable to determine themaximum of the classical Stirling numbers Here we derive aformula that determines the value of the index correspondingto the maximum of the signless Stirling numbers of the firstkind

The signless Stirling numbers of the first kind [6] are thesum of all products of 119899 minus 119896 different integers taken from1 2 3 119899 minus 1 That is

|119904 (119899 119896)| = sum1le1198941lt1198942ltsdotsdotsdotlt119894119899minus119896le119899minus1

11989411198942sdot sdot sdot 119894119899minus119896

(63)


Table 2 Values of 11990410(119899 119896)

119899119896 0 1 2 30 11 10 12 90 19 13 720 242 27 14 5040 2414 431 345 30240 19524 5000 6356 151200 127860 44524 81757 604800 662640 305956 772248 1814400 2592720 1580508 5376289 3628800 6999840 5753736 265576410 3628800 10628640 12753576 8409500

Table 3 Values of |119904(119899 119896)| for 0 le 119899 le 10

119904(119899 119896) 0 1 2 3 4 50 11 0 12 0 1 13 0 2 3 14 0 6 11 6 15 0 24 50 35 10 16 0 120 274 225 85 157 0 720 1764 1624 735 1758 0 540 13068 13132 6769 19609 0 40320 109584 118124 67284 2244910 0 362880 1026576 1172700 723680 269325

Using Theorem 10 |119904(119899 119896)| = sum119899minus1119899minus119896

We use 119896119899to denote

the largest value of 119896 for which sum119899minus1119899minus119896

is maximum With1199061= 1 119906

2= 2 119906

119899minus1= 119899 minus 1 we have

119896119899= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

1

1198942(1 +

1

119894)minus1

+ 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

(1

119894minus

1

119894 + 1) + 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894 + 1+ 119900 (1)]

(64)

Using (53) we see that

119899minus1

sum119894=1

1

1 + 119894= log 119899 minus log 1 + 120574 + 119900 (1) (65)

Therefore we have

119896119899= [log 119899 + 120574 + 119900 (1)] (66)

Example 15 It is shown in Table 3 that the maximum valueof |119904(119899 119896)| when 119899 = 7 occurs at 119896 = 2 Using (66) it

can be verified that the maximum element of the sequence|119904(7 119896)|

7

119896=0occurs at

1198967= [log 7 + 120574 + 119900 (1)]

= [195 + 05772 + 119900 (1)]

= [253 + 119900 (1)]

asymp 2

(67)

Moreover when 119899 = 10 the maximum value occurs at

11989610

= [log 10 + 120574 + 119900 (1)]

= [230 + 05772 + 119900 (1)]

= [28772 + 119900 (1)]

asymp 3

(68)

Recently a paper by Cakic et al [7] established explicitformulas for multiparameter noncentral Stirling numberswhich are expressible in symmetric function forms One maythen try to investigate the location of the maximum value ofthese numbers using the Erdos-Stone theorem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors wish to thank the referees for reading the paperthoroughly

References

[1] M Koutras ldquoNoncentral Stirling numbers and some applica-tionsrdquo Discrete Mathematics vol 42 no 1 pp 73ndash89 1982

[2] I Mezo ldquoOn the maximum of 119903-Stirling numbersrdquo Advances inApplied Mathematics vol 41 no 3 pp 293ndash306 2008

[3] P Erdos ldquoOn a conjecture of Hammersleyrdquo Journal of theLondon Mathematical Society vol 28 pp 232ndash236 1953

[4] A de Medicis and P Leroux ldquoGeneralized Stirling numbersconvolution formulae and 119901 119902-analoguesrdquo Canadian Journal ofMathematics vol 47 no 3 pp 474ndash499 1995

[5] E H Lieb ldquoConcavity properties and a generating function forStirling numbersrdquo Journal of CombinatorialTheory vol 5 no 2pp 203ndash206 1968

[6] L Comtet Advanced Combinatorics Reidel Dordrecht TheNetherlands 1974

[7] N P Cakic B S El-Desouky and G V Milovanovic ldquoExplicitformulas and combinatorial identities for generalized StirlingnumbersrdquoMediterranean Journal of Mathematics vol 10 no 1pp 57ndash72 2013

Research ArticleA Domain-Specific Architecture for ElementaryFunction Evaluation

Anuroop Sharma and Christopher Kumar Anand

Department of Computing and Software McMaster University Hamilton ON Canada L8S 4K1

Correspondence should be addressed to Christopher Kumar Anand anandcmcmasterca

Received 25 December 2014 Accepted 3 August 2015


Copyright copy 2015 A Sharma and C K AnandThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We propose a Domain-Specific Architecture for elementary function computation to improve throughput while reducing powerconsumption as a model for more general applications support fine-grained parallelism by eliminating branches and eliminate theduplication required by coprocessors by decomposing computation into instructions which fit existing pipelined execution modelsand standard register files Our example instruction architecture (ISA) extension supports scalar and vectorSIMD implementationsof table-based methods of calculating all common special functions with the aim of improving throughput by (1) eliminating theneed for tables in memory (2) eliminating all branches for special cases and (3) reducing the total number of instructions Twonew instructions are required a table lookup instruction and an extended-precision floating-point multiply-add instruction withspecial treatment for exceptional inputs To estimate the performance impact of these instructions we implemented them in amodified CellBE SPU simulator and observed an average throughput improvement of 25 times for optimized loops mappingsingle functions over long vectors

1 Introduction

Elementary function libraries are often called from perfor-mance-critical code sections and hence contribute greatly tothe efficiency of numerical applications and the performanceof these and libraries for linear algebra largely determinethe performance of important applications Current hardwaretrends impact this performance as

(i) longer pipelines and wider superscalar dispatchfavour implementations which distribute computa-tion across different execution units and present thecompiler with more opportunities for parallel execu-tion but make branches more expensive

(ii) Single-Instruction-Multiple-Data (SIMD)parallelismmakes handling cases via branches very expensive

(iii) memory throughput and latency which are notadvancing as fast as computational throughput hinderthe use of lookup tables

(iv) power constraints limit performance more than area

The last point is interesting and gives rise to the notion ofldquodark siliconrdquo in which circuits are designed to be un- orunderused to save power The consequences of these thermallimitations versus silicon usage have been analyzed [1] anda number of performance-stretching approaches have beenproposed [2] including the integration of specialized copro-cessors

Our proposal is less radical instead of adding special-ized coprocessors add novel fully pipelined instructions toexisting CPUs and GPUs use the existing register file reuseexisting silicon for expensive operations for example fusedmultiply-add operations and eliminate costly branches butadd embedded lookup tables which are a very effective useof dark silicon In the present paper we demonstrate thisapproach for elementary function evaluation that is libmfunctions and especially vector versions of them

To optimize performance our approach takes the suc-cessful accurate table approach of Gal et al [3 4] coupledwith algorithmic optimizations [5 6] and branch and tableunifications [7] to reduce all fixed-power- logarithm- andexponential-family functions to a hardware-based lookup



followed by a handful of floating-point operations mostlyfused multiply-add instructions evaluating a single polyno-mial Other functions like pow require multiple such basicstages but no functions require branches to handle excep-tional cases including subnormal and infinite values

Although fixed powers (including square roots and recip-rocals) of most finite inputs can be efficiently computed usingNewton-Raphson iteration following a software or hardwareestimate [8] such iterations necessarily introduce NaN inter-mediate values which can only be corrected using additionalinstructions (branches predication or selection) Thereforeour proposed implementations avoid iterative methods

For evaluation of the approach the proposed instructionswere implemented in a CellBE [9] SPU simulator and algo-rithms for a standard math function library were developedthat leverage these proposed additionsOverall we found thatthe new instructions would result in an average 25 timesthroughput improvement on this architecture versus currentpublished performance results (Mathematical AccelerationSubsystem 50 IBM) Given the simple data dependencygraphs involved we expect similar improvements fromimplementing these instructions on all two-way SIMD archi-tectures supporting fused multiply-add instructions Higher-way SIMD architectures would likely benefit more

In the following the main approach is developed andthe construction of two representative functions log119909 andlog(119909 + 1) is given in detail providing insight by exampleinto the nature of the implementation In some sense theserepresent the hardest case although the trigonometric func-tions require multiple tables and there is some computationof the lookup keys the hardware instructions themselves aresimpler For a complete specification of the algorithms usedsee [10]

2 New Instructions

Driven by hardware floating-point instructions the adventof software pipelining and shortening of pipeline stagesfavoured iterative algorithms (see eg [11]) the long-runningtrend towards parallelism has engendered a search for sharedexecution units [12] and in a more general sense a focus onthroughput rather than low latency In terms of algorithmicdevelopment for elementary functions thismakes combiningshort-latency seed or table value lookups with standardfloating-point operations attractive exposing the whole com-putation to software pipelining by the scheduler

In proposing Instruction Set Architecture (ISA) exten-sions one must consider four constraints

(i) the limit on the number of instructions imposed bythe size of the machine word and the desire for fast(ie simple) instruction decoding

(ii) the limit on arguments and results imposed by thearchitected number of ports on the register file

(iii) the limit on total latency required to prevent anincrease in maximum pipeline depth

(iv) the need to balance increased functionality withincreased area and power usage

x

f(x)

i

(ii) and (iii)1

c

middot middot middotfmaX fma fma

fma or fmlookup fn 1 lookup fn 2

Figure 1 Data flow graph with instructions on vertices for log119909roots and reciprocals Most functions follow the same basic patternor are a composition of such patterns

As new lithography methods cause processor sizes to shrinkthe relative cost of increasing core area for new instructions isreducedThenet costmay even be negative if the new instruc-tions can reduce code anddata size thereby reducing pressureon the memory interface (which is more difficult to scale)

To achieve a performance benefit ISA extensions shoulddo one or more of the following

(i) reduce the number of machine instructions in com-piled code

(ii) move computation away from bottleneck executionunits or dispatch queues or

(iii) reduce register pressure

Considering the above limitations and ideals we proposeadding two instructions the motivation for which followsbelow

d = fmaX a b c an extended-range floating-pointmultiply-add with the first argument having 12 expo-nent bits and 51 mantissa bits and nonstandardexception handlingt1 = lookup a b f t an enhanced table lookup basedon one or two arguments and containing immediateargument specifying the function number and thesequence of the lookup for example the first lookupused for range reduction or the second lookup usedfor function reconstruction

It is easiest to see them used in an example Figure 1describes the data flow graph (omitting register constants)which is identical for a majority of the elementary functionsThe correct lookup is specified as an immediate argumentto lookup the final operation being fma for the log func-tions and fm otherwise All of the floating-point instructionsalso take constant arguments which are not shown Forexample the fmaX takes an argument which is minus1

The dotted box in Figure 1 represents a varying number offused multiply-adds used to evaluate a polynomial after themultiplicative range reduction performed by the fmaX Themost common size for these polynomials is order three sothe result of the polynomial (the left branch) is available fourfloating-point operations later (typically about 24ndash28 cycles)than the result 1119888 The second lookup instruction performsa second lookup for example the log 119909 looks up log

2119888 and

substitutes exceptional results (plusmninfin NaN) when necessaryThe final fma or fm instruction combines the polynomialapproximation on the reduced interval with the table value


The gray lines indicate two possible data flows for threepossible implementations

(i) the second lookup instruction is a second lookupusing the same input

(ii) the second lookup instruction retrieves a value savedby the first lookup (in final or intermediate form)from a FIFO queue

(iii) the second lookup instruction retrieves a value savedin a slot according to an immediate tag which is alsopresent in the corresponding first lookup

In the first case the dependency is direct In the second twocases the dependency is indirect via registers internal to theexecution unit handling the lookups

All instruction variations have two register inputs andone or no outputs so they will be compatible with existingin-flight instruction and register tracking On lean in-orderarchitectures the variants with indirect dependenciesmdash(ii)and (iii)mdashreduce register pressure and simplify modulo loopscheduling This would be most effective in dedicated com-putational cores like the SPUs in which preemptive contextswitching is restricted

The variant (iii) requires additional instruction decodelogic but may be preferred over (ii) because tags allowlookup instructions to execute in different orders and forwide superscalar processors the tags can be used by the unitassignment logic to ensure that matching lookup instruc-tions are routed to the same units On Very Long InstructionWord machines the position of lookups could replace oraugment the tag

In low-power environments the known long minimumlatency between the lookups would enable hardware design-ers to use lower power but longer latency implementations ofmost of the second lookup instructions

To facilitate scheduling it is recommended that the FIFOor tag set be sized to the power of two greater than or equalto the latency of a floating-point operation In this casethe number of registers required will be less than twice theunrolling factor which is much lower than what is possiblefor code generated without access to such instructions Thecombination of small instruction counts and reduced registerpressure eliminate the obstacles to inlining of these functions

We recommend that lookups be handled by either aloadstore unit or for vector implementations with a com-plex integer unit by that unit This code will be limited byfloating-point instruction dispatch so moving computationout of this unit will increase performance

21 Exceptional Values A key advantage of the proposednew instructions is that the complications associated withexceptional values (0 infin NaN and values with over- orunderflow at intermediate stages) are internal to the instruc-tions eliminating branches and predicated execution

Iterative methods with table-based seed values cannotachieve this in most cases because

(1) in 0 and plusmninfin cases the iteration multiplies 0 by infinproducing a NaN

(2) to prevent overunderflow for high and low inputexponents matched adjustments are required beforeand after polynomial evaluation or iterations

By using the table-based instruction twice once to look upthe value used in range reduction and once to look up thevalue of the function corresponding to the reduction andintroducing an extended-range floating-point representationwith special handling for exceptions we can handle bothtypes of exceptions without extra instructions

In the case of finite inputs the value 2minus119890119888 such that

2minus119890

119888sdot 119909 minus 1 isin [minus2

minus119873

2minus119873

] (1)

returned by the first lookup is a normal extended-rangevalue In the case of subnormal inputs extended-range arerequired to represent this lookup value because normal IEEEvalue would saturate to infin Treatment of the large inputswhich produce IEEE subnormal as their approximate recip-rocals can be handled (similar to normal inputs) using theextended-range representation The extended-range numberis biased by +2047 and the top binary value (4095) isreserved for plusmninfin and NaNs and 0 is reserved for plusmn0 similarto IEEE floating point When these values are supplied asthe first argument of fmaX they override the normal valuesand fmaX simply returns the corresponding IEEE bit pattern

The second lookup instruction returns an IEEE doubleexcept when used for divide in which case it also returns anextended-range result

In Table 2 we summarize how each case is handled anddescribe it in detail in the following section Each cell inTable 2 contains the value used in the reduction followedby the corresponding function value The first is given as anextended-range floating-point number which trades one bitof stored precision in the mantissa with a doubling of theexponent range In all cases arising in this library the extra bitwould be one of several zero bits so no actual loss of precisionoccurs For the purposes of elementary function evaluationsubnormal extended-range floating-point numbers are notneeded so they do not need to be supported in the floating-point execution unit As a result themodifications to supportextended-range numbers as inputs are minor

Take for example the first cell in the table recip com-puting 1119909 for a normal positive input Although the abstractvalues are both 2minus119890119888 the bit patterns for the two lookupsare different meaning that 1119888must be representable in bothformats In the next cell however for some subnormal inputs2minus119890

119888 is representable in the extended range but not in IEEEfloating point because the addition of subnormal numbersmakes the exponent range asymmetrical As a result thesecond value may be saturated toinfin The remaining cells inthis row show that for plusmninfin input we return 0 from bothlookups but for plusmn0 inputs the first lookup returns 0 andthe second lookup returns plusmninfin In the last column we seethat for negative inputs the returned values change the signThis ensures that intermediate values are always positive andallows the polynomial approximation to be optimized togive correctly rounded results on more boundary cases Bothlookups return quiet NaN outputs for NaN inputs


Contrast this with the handling of approximate reciprocalinstructions For the instructions to be useful as approxi-mations 0 inputs should returninfin approximations and viceversa but if the approximation is improved using Newton-Raphson then the multiplication of the input by the approx-imation produces a NaN which propagates to the final result

The other cases are similar in treating 0 and infin inputsspecially Noteworthy variations are that log

2119909 multiplica-

tively shifts subnormal inputs into the normal range so thatthe normal approximation can be used and then additivelyshifts the result of the second lookup to compensate and 2119909returns 0 and 1 for subnormal inputs because the polynomialapproximation produces the correct result for the wholesubnormal range

In Table 2 we list the handling of exceptional casesAll exceptional values detected in the first argument areconverted to the IEEE equivalent and are returned as theoutput of the fmaX as indicated by subscript

119891(for final)The

subscripted NaNs are special bit patterns required to producethe special outputs needed for exceptional cases For examplewhen fmaX is executed with NaN

1as the first argument (one

of the multiplicands) and the other two arguments are finiteIEEE values the result is 2 (in IEEE floating-point format)

fmaX NaN1finite1finite2= NaN

1sdot finite

1+ finite

2= 2 (2)

If the result of multiplication is an infin and the addend isinfin with the opposite sign then the result is zero althoughnormally it would be a NaN If the addend is a NaN thenthe result is zero For the other values indicated by ldquo119888rdquo inTable 2 fmaX operates as a usual fused multiply-accumulateexcept that the first argument (amultiplicand) is an extended-range floating-point number For example the fused multi-plication and addition of finite arguments saturate to plusmninfin inthe usual way

Finally for exponential functions which return fixedfinite values for a wide range of inputs (including infinities)it is necessary to override the range reduction so that itproduces an output which results in a constant value afterthe polynomial approximation In the case of exponentialany finite value which results in a nonzero polynomial valuewill do because the second lookup instruction returns 0 orinfin and multiplication by any finite value will return 0 asrequired

Lookup Internals The lookup instructions perform similaroperations for each of the elementary functions we have con-sidered The function number is an immediate argument Inassembly language each function could be a different macrowhile in high level languages the pair could be represented bya single function returning a tuple or struct

A simplified data flow for the most complicated caselog2119909 is represented in Figure 2 The simplification is the

elimination of the many single-bit operations necessary tokeep track of exceptional conditions while the operationsto substitute special values are still shown Critically thediagram demonstrates that the operations around the corelookup operations are all of low complexity The graph isexplained in the following where letter labels correspondto the blue colored labels in Figure 2 This representation is

for variant (ii) or (iii) for the second lookup and includes adotted line on the centre-right of the figure at (a) delineatinga possible set of values to save at the end of the first lookupwhere the part of the data flow below the line is computed inthe second lookup instruction

Starting from the top of the graph the input (b) is used togenerate two values (c) and (d) 2minus119890120583 and 119890+log

2120583 in the case

of log2119909 The heart of the operation is two lookup operations

(e) and (f) with a common index In implementation (i)the lookups would be implemented separately while in theshared implementations (ii) and (iii) the lookups could beimplemented more efficiently together

Partial decoding of subnormal inputs (g) is required forall of the functions except the exponential functions Only theleading nonzero bits are needed for subnormal values andonly the leading bits are needed for normal values but thenumber of leading zero bits (h) is required to properly formthe exponent for themultiplicative reductionTheonly switch(i) needed for the first lookup output switches between thereciprocal exponents valid in the normal and subnormalcases respectively Accurate range reduction for subnormalrequires both extreme end points for example 12 and 1because these values are exactly representable As a result twoexponent values are required and we accommodate this bystoring an exponent bit (j) in addition to the 51 mantissa bits

On the right hand side the lookup (e) for the secondlookup operation also looks up a 4-bit rotation which alsoserves as a flag We need 4 bits because the table size 212implies that we may have a variation in the exponent of theleading nonzero bit of up to 11 for nonzero table values Thisallows us to encode in 30 bits the floating mantissa used toconstruct the second lookup output This table will alwayscontain a 0 and is encoded as a 12 in the bitRot field Inall other cases the next operation concatenates the implied1 for this floating-point format This gives us an effective 31bits of significance (l) which is then rotated into the correctposition in a 42-bit fixed point number Only the high-orderbit overlaps the integer part of the answer generated from theexponent bits so this value needs to be padded Because theoutput is an IEEE float the contribution of the (padded) valueto the mantissa of the output will depend on the sign of theinteger exponent part This sign is computed by adding 1 (m)to the biased exponent in which case the high-order bit is 1if and only if the exponent is positive This bit (n) is used tocontrol the sign reversal of the integer part (o) and the sign ofthe sign reversal of the fractional part which is optimized bypadding (p) after xoring (q) but before the +1 (r) required tonegate a tworsquos-complement integer

The integer part has now been computed for normalinputs but we need to switch (s) in the value for subnormalinputs which we obtain by biasing the number of leadingzeros computed as part of the first step The apparent 75-bitadd (t) is really only 11 bits with 10 of the bits coming frompadding on one side This fixed-point number may containleading zeros but the maximum number is log

2((maximum

integer part) minus (smallest nonzero table value)) = 22 for thetested table size As a result the normalization (u) only needsto check for up to 22 leading zero bits and if it detectsthat number set a flag to substitute a zero for the exponent


Inputsign exp mant

Drop leading zeros

numZeros + 1

lowExpBit0b1111111111

Add

Switch

s e12 m51

lookup output

s e11 m52retrieve output

exp

Add 1

expP1

adjustedE

Switch

bitRot

Rotate pad 10 left by

Add 0x7ff

Add 0x3ff

Switch

Concatenate bit

xor

Add drop carry

moreThan22Zeros

isNot12

Concatenate bit

51

42

42 + 1

44

30

31

52

52

52

52

22Complement bitsBits in data path

Add

Rotate bits either to clear leading zeros or according to second (length) argument

Select one of two inputs or immediate according to logical input (not shown for exceptions)

(c)

(b)

(d)

(e)

(f)

(g)

Recommended lookupretrieve boundary requiring 46-bit storage

(h)

(i)

(j)

(k)

leading12Mant

(a)

(p)

(r)

(o)

(l)

(m)

(n)

(q)

(s)

(t)

(u)

(v)

5

sim

sim

0x7ffSubs

Subs 0b0 0

Subtract from0x3ff + 10

Drop up to 22 leadingzeros and first 1 round

Add left justifieddrop carry

12-bit lookup

12-bit lookup

ge1

e(last 10bits)lt1

sim

Addsubtract right-justified inputs unless stated

0b10 0Subs

Subs 0b0 0

low 10bits4

6

11

11

11

11

12

Lower 10

Figure 2 Bit flow graph with operations on vertices for log 119909 lookup Shape indicates operation type and line width indicates data pathswidth in bits Explanation of function in the text


Table 1 Accuracy throughput and table size (for SPUdouble precision)

Function Cyclesdoublenew

CyclesdoubleSPU Speedup () Max error (ulps) Table size (119873) Poly order log

2119872

recip 3 113 376 0500 2048 3 infin

div 35 149 425 1333 recip 3sqrt 3 154 513 0500 4096 3 18rsqrt 3 146 486 0503 4096 3 infin

cbrt 83 133 160 0500 8192 3 18rcbrt 10 161 161 0501 rcbrt 3 infin

qdrt 75 276 368 0500 8192 3 18rqdrt 83 196 229 0501 rqdrt 3 18log2 25 146 584 0500 4096 3 18log21p 35 na na 1106 log2 3log 35 138 394 1184 log2 3log1p 45 225 500 1726 log2 3exp2 45 130 288 1791 256 4 18exp2m1 55 na na 129 exp2 4exp 50 144 288 155 exp2 4expm1 55 195 354 180 exp2 4atan2 75 234 311 0955 4096 2 18atan 75 185 246 0955 atan2 2 + 3asin 11 272 247 1706 atan2 2 + 3 + 3acos 11 271 246 0790 atan2 2 + 3 + 3sin 11 166 150 1474 128 3 + 3 52cos 10 153 153 1025 sin 3 + 3tan 245 276 113 2051 sin 3 + 3 + 3

(v) (the mantissa is automatically zero) The final switchessubstitute special values for plusmninfin and a quiet NaN

If the variants (ii) and (iii) are implemented either thehidden registers must be saved on contextcore switches orsuch switches must be disabled during execution of theseinstructions or execution of these instructions must belimited to one thread at a time

3 Evaluation

Two types of simulations of these instructions were carriedout First to test accuracy our existing CellBE functionalinterpreter see [13] was extended to include the new instruc-tions Second we simulated the log lookups and fmaX usinglogical operations on bits that is a hardware simulationwithout timing and verified that the results match theinterpreter

Performance Since the dependency graphs (as typified byFigure 1) are close to linear and therefore easy to sched-ule optimally the throughput and latency of software-pipelined loops will be essentially proportional to the num-ber of floating-point instructions Table 1 lists the expectedthroughput for vector math functions with and withoutthe new instructions Figure 3 demonstrated the relativemeasured performance improvements Overall the addition

of hardware instructions to the SPU results in a mean 25timesthroughput improvement for the whole function libraryPerformance improvements on other architectures will varybut would be similar since the acceleration is primarily theresult of eliminating instructions for handling exceptionalcases

Accuracy We tested each of the functions by simulatingexecution for 20000 pseudorandom inputs over their naturalranges or [minus1000120587 1000120587] for trigonometric functions andcomparing each value to a 500-digit-precision Maple [14]reference Table 1 shows a maximum 2051 ulp error withmany functions close to correct rounding This is well withintheOpenCL specifications [15] and shows very good accuracyfor functions designed primarily for high throughput andsmall code size For applications requiring even higher accu-racy larger tables could be used and polynomials with betterrounding properties could be searched for using the lattice-basis-reduction method of [16]

4 Conclusion

We have demonstrated considerable performance improve-ments for fixed power exponential and logarithm calcula-tions by using novel table lookup and fused multiply-add


Table 2 (a) Values returned by lookup instructions for IEEE floating-point inputs (minus1)1199042119890119891 which rounds to the nearest integer 119868 =rnd((minus1)1199042119890119891) In case of exp2 inputs ltminus1074 are treated as minusinfin and inputs gt1024 are treated asinfin For inputs ltminus1022 we create subnormalnumbers for the second lookup (b c) Treatment of exceptional values by fmaX follows from that of addition and multiplication The firstargument is given by the row and the second by the column Conventional treatment is indicated by a ldquo119888rdquo and unusual handling by specificconstant values

(a)

Function Finite gt 0 +infin minusinfin plusmn0 Finite lt 0

recip (2minus119890

119888)ext (2minus119890

119888)sat 0 0 0 0 0 plusmninfin (minus2minus119890

119888)ext (minus2minus119890

119888)sat

sqrt (2minus119890

119888)ext 21198902

119888 0infin 0 NaN 0 0 0 NaN

rsqrt (2minus119890119888)ext 2minus1198902119888 0 0 0 NaN 0infin 0 NaN

log2 (2minus119890119888)ext 119890 + log2119888 0infin 0 NaN 0 minusinfin 0 NaN

exp2 119888 2119868 sdot 2119888 0infin NaN 0 0 1 119888 2minus119868 sdot 2119888

(b)

+ext Finite minusinfin infin NaNFinite 119888 119888 119888 0

minusinfin 119888 119888 0 0

infin 119888 0 119888 0

NaN 119888 119888 119888 0

(c)

lowastext Finite minusinfin infin NaNFinite = 0 119888 2 2 2minusinfin minusinfin

119891minusinfin119891

minusinfin119891

minusinfin119891

infin infin119891

infin119891

infin119891

infin119891

NaN0

NaN119891

NaN119891

NaN119891

NaN119891

NaN1

2119891

2119891

2119891

2119891

NaN2

1radic2119891

1radic2119891

1radic2119891

1radic2119891

NaN3

0119891

0119891

0119891

0119891

recipdiv

sqrtrsqrtcbrt

rcbrtqdrt

rqdrtlog2

loglog1pexp2

expexpm1

atan2atanasinacos

sincostan

0 5 10 15 20 25 30

Figure 3 Throughput measured in cycles per double for imple-mentations of elementary function with (upper bars) and without(lower bars) the novel instructions proposed in this paper

instructions in simple branch-free accurate-table-based algo-rithms Performance improved less for trigonometric func-tions but this improvement will grow with more coresandor wider SIMD These calculations ignored the effect ofreduced power consumption caused by reducing instructiondispatch and function calls and branching and reducingmemory accesses for large tables which will mean that thesealgorithms will continue to scale longer than conventionalones

For target applications just three added opcodes pack alot of performance improvement but designing the instruc-tions required insights into the algorithms and even a newalgorithm [7] for the calculation of these functionsWe inviteexperts in areas such as cryptography and data compressionto try a similar approach




Acknowledgments

The authors thank NSERC MITACS Optimal Computa-tional Algorithms Inc and IBM Canada for financial sup-port Some work in this paper is covered by US Patent6804546

References

[1] H Esmaeilzadeh E Blem R St Amant K Sankaralingam andD Burger ldquoDark silicon and the end ofmulticore scalingrdquoACMSIGARCH Computer Architecture News vol 39 no 3 pp 365ndash376 2011

[2] M B Taylor ldquoIs dark silicon useful Harnessing the fourhorsemen of the coming dark silicon apocalypserdquo inProceedingsof the 49th Annual Design Automation Conference (DAC rsquo12) pp1131ndash1136 ACM San Francisco Calif USA 2012

[3] S Gal ldquoComputing elementary functions a new approachfor achieving high accuracy and good performancerdquo in Accu-rate Scientific Computations Symposium Bad Neuenahr FRGMarch 12-14 1985 Proceedings vol 235 of Lecture Notes inComputer Science pp 1ndash16 Springer London UK 1986

[4] S Gal and B Bachelis ldquoAccurate elementary mathematicallibrary for the IEEE floating point standardrdquoACMTransactionson Mathematical Software vol 17 no 1 pp 26ndash45 1991

[5] P T P Tang ldquoTable-driven implementation of the logarithmfunction in IEEE floating-point arithmeticrdquo ACM Transactionson Mathematical Software vol 16 no 4 pp 378ndash400 1990

[6] P T P Tang ldquoTable-driven implementation of the Expm 1 func-tion in IEEE floating-point arithmeticrdquo ACM Transactions onMathematical Software vol 18 no 2 pp 211ndash222 1992

[7] C K Anand and A Sharma ldquoUnified tables for exponentialand logarithm familiesrdquo ACM Transactions on MathematicalSoftware vol 37 no 3 article 28 2010

[8] C T Fike ldquoStarting approximations for square root calculationon IBM system 360rdquo Communications of the ACM vol 9 no4 pp 297ndash299 1966

[9] IBM Corporation Cell Broadband Engine Programming Hand-book IBM Systems and Technology Group 2008 httpswww-01ibmcomchipstechlibtechlibnsfproductfamiliesPowerPC

[10] A Sharma Elementary function evaluation using new hardwareinstructions [MS thesis] McMaster University 2010

[11] M S Schmookler R C Agarwal and F G Gustavson ldquoSeriesapproximation methods for divide and square root in thePower3TM processorrdquo inProceedings of the 14th IEEE Symposiumon Computer Arithmetic (ARITH rsquo99) pp 116ndash123 IEEE Com-puter Society Adelaide Australia 1999

[12] M D Ercegovac and T Lang ldquoMultiplicationdivisionsquareroot module for massively parallel computersrdquo Integration theVLSI Journal vol 16 no 3 pp 221ndash234 1993

[13] C K Anand and W Kahl ldquoAn optimized Cell BE specialfunction library generated by Coconutrdquo IEEE Transactions onComputers vol 58 no 8 pp 1126ndash1138 2009

[14] MaplesoftMaplesoft Maple 12 User Manual 2008[15] KOWGroupTheOpenCLSpecificationVersion 10Document

Revision 29 2008[16] N Brisebarre and S Chevillard ldquoEfficient polynomial L1

approximationsrdquo in Proceedings of the 18th IEEE Symposiumon Computer Arithmetic pp 169ndash176 IEEE Computer SocietyPress Los Alamitos Calif USA 2007

Research ArticleExplicit Formulas for Meixner Polynomials

Dmitry V Kruchinin12 and Yuriy V Shablya1

1Tomsk State University of Control Systems and Radioelectronics 40 Lenina Avenue Tomsk 634050 Russia2National Research Tomsk Polytechnic University 30 Lenin Avenue Tomsk 634050 Russia

Correspondence should be addressed to Dmitry V Kruchinin kruchinindmgmailcom

Received 5 September 2014 Accepted 26 December 2014

Academic Editor Hari M Srivastava

Copyright copy 2015 D V Kruchinin and Y V Shablya This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Using notions of composita and composition of generating functions we show an easy way to obtain explicit formulas for somecurrent polynomials Particularly we consider the Meixner polynomials of the first and second kinds

1 Introduction

There are many authors who have studied polynomials andtheir properties (see [1ndash10]) The polynomials are applied inmany areas ofmathematics for instance continued fractionsoperator theory analytic functions interpolation approxi-mation theory numerical analysis electrostatics statisticalquantummechanics special functions number theory com-binatorics stochastic processes sorting and data compres-sion

The research area of obtaining explicit formulas forpolynomials has received much attention from Srivastava[11 12] Cenkci [13] Boyadzhiev [14] and Kruchinin [15ndash17]

The main purpose of this paper is to obtain explicitformulas for the Meixner polynomials of the first and secondkinds

In this paper we use a method based on a notion ofcomposita which was presented in [18]

Definition 1 Suppose 119865(119905) = sum119899gt0

119891(119899)119905119899 is the generating

function in which there is no free term 119891(0) = 0 From thisgenerating function we can write the following condition

[119865 (119905)]119896

= sum119899gt0

119865 (119899 119896) 119905119899

(1)

The expression 119865(119899 119896) is composita [19] and it is denotedby 119865Δ(119899 119896) Below we show some required rules and opera-tions with compositae

Theorem 2 Suppose 119865(119905) = sum119899gt0


function and 119865Δ(119899 119896) is composita of 119865(119905) and 120572 is constantFor the generating function 119860(119905) = 120572119865(119905) composita is equal to

119860Δ

(119899 119896) = 120572119896

119865Δ

(119899 119896) (2)


119891(119899)119905119899 is the generatingfunction and 119865Δ(119899 119896) is composita of 119865(119905) and 120572 is constantFor the generating function 119860(119905) = 119865(120572119905) composita is equal to

119860Δ

(119899 119896) = 120572119899

119865Δ

(119899 119896) (3)


119891(119899)119905119899 119877(119905) = sum119899gt0

119903(119899)119905119899

are generating functions and 119865Δ(119899 119896) 119877Δ(119899 119896) are theircompositae Then for the composition of generating functions119860(119905) = 119877(119865(119905)) composita is equal to

119860Δ

(119899 119896) =

119899

sum119898=119896

119865Δ

(119899119898)119866Δ

(119898 119896) (4)


119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899

are generating functions and 119865Δ(119899 119896) is composita of 119865(119905)Then for the composition of generating functions 119860(119905) =

119877(119865(119905)) coefficients of generating functions 119860(119905) = sum119899ge0

119886(119899)119905119899

are

119886 (119899) =

119899

sum119896=1

119865Δ

(119899 119896) 119903 (119896)

119886 (0) = 119903 (0)

(5)




119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899

are generating functions Then for the product of generatingfunctions 119860(119905) = 119865(119905)119877(119905) coefficients of generating functions119860(119905) = sum

119899ge0119886(119899)119905119899 are

119886 (119899) =

119899

sum119896=0

119891 (119896) 119903 (119899 minus 119896) (6)

In this paper we consider an application of this math-ematical tool for the Bessel polynomials and the Meixnerpolynomials of the first and second kinds

2 Bessel Polynomials

Krall and Frink [20] considered a new class of polynomialsSince the polynomials connected with the Bessel functionthey called them the Bessel polynomialsThe explicit formulafor the Bessel polynomials is

119910119899(119909) =

119899

sum119896=0

(119899 + 119896)

(119899 minus 119896)119896(119909

2)119896

(7)

ThenCarlitz [21] defined a class of polynomials associatedwith the Bessel polynomials by

119901119899(119909) = 119909

119899

119910119899minus1

(1

119909) (8)

The 119901119899(119909) is defined by the explicit formula [22]

119901119899(119909) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)(1

2)119899minus119896

119909119896 (9)

and by the following generating functioninfin

sum119899=0

119901119899(119909)

119899119905119899

= 119890119909(1minusradic1minus2119905)

(10)

Using the notion of composita we can obtain an explicitformula (9) from the generating function (10)

For the generating function 119862(119905) = (1 minus radic1 minus 4119905)2

composita is given as follows (see [19])

119862Δ

(119899 119896) =119896

119899(2119899 minus 119896 minus 1

119899 minus 1) (11)

We represent 119890119909(1minusradic1minus2119905) as the composition of generatingfunctions 119860(119861(119905)) where

119860 (119905) = 119909 (1 minus radic1 minus 2119905) = 2119909119862(119905

2)

119861 (119905) = 119890119905

=

infin

sum119899=0

1

119899119905119899

(12)

Then using rules (2) and (3) we obtain composita of119860(119905)

119860Δ

(119899 119896) = (2119909)119896

(1

2)119899

119862Δ

(119899 119896) = 2119896minus119899

119909119896119896

119899(2119899 minus 119896 minus 1

119899 minus 1)

(13)

Coefficients of the generating function119861(119905) = suminfin

119899=0119887(119899)119905119899

are equal to

119887 (119899) =1

119899 (14)

Then using (5) we obtain the explicit formula

119901119899(119909) = 119899

119899

sum119896=1

119860Δ

(119899 119896) 119887 (119896) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)2119896minus119899

119909119896

(15)

which coincides with the explicit formula (9)

3 Meixner Polynomials of the First Kind

TheMeixner polynomials of the first kind are defined by thefollowing recurrence relation [23 24]

119898119899+1

(119909 120573 119888) =(119888 minus 1)

119899

119888119899

sdot ((119909 minus 119887119899)119898119899(119909 120573 119888) minus 120582

119899119898119899minus1

(119909 120573 119888))

(16)

where

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) = 119909 minus 119887

0

119887119899=

(1 + 119888) 119899 + 120573119888

1 minus 119888

120582119899=

119888119899 (119899 + 120573 minus 1)

(1 minus 119888)2

(17)

Using (16) we obtain the first few Meixner polynomialsof the first kind

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) =

119909 (119888 minus 1) + 120573119888

119888

1198982(119909 120573 119888)

=1199092 (119888 minus 1)

2

+ 119909 ((2120573 + 1) 1198882 minus 2120573119888 minus 1) + (120573 + 1) 1205731198882

1198882

(18)

The Meixner polynomials of the first kind are defined bythe following generating function [22]

infin

sum119899=0

119898119899(119909 120573 119888)

119899119905119899

= (1 minus119905

119888)119909

(1 minus 119905)minus119909minus120573

(19)

Using the notion of composita we can obtain an explicitformula119898

119899(119909 120573 119888) from the generating function (19)


First we represent the generating function (19) as a prod-uct of generating functions 119860(119905) 119861(119905) where the functions119860(119905) and 119861(119905) are expanded by binomial theorem

119860 (119905) = (1 minus119905

119888)119909

=

infin

sum119899=0

(119909

119899) (minus1)

119899

119888minus119899

119905119899

119861 (119905) = (1 minus 119905)minus119909minus120573

=

infin

sum119899=0

(minus119909 minus 120573

119899) (minus1)

119899

119905119899

(20)

Coefficients of the generating functions 119860(119905) =

suminfin

119899=0119886(119899)119905119899 and 119861(119905) = sum

infin

119899=0119887(119899)119905119899 are respectively given as

follows

119886 (119899) = (119909

119899) (minus1)

119899

119888minus119899

119887 (119899) = (minus119909 minus 120573

119899) (minus1)

119899

(21)

Then using (6) we obtain a new explicit formula for theMeixner polynomials of the first kind

119898119899(119909 120573 119888) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896)

= (minus1)119899

119899

119899

sum119896=0

(119909

119896)(

minus119909 minus 120573

119899 minus 119896) 119888minus119896

(22)

4 Meixner Polynomials of the Second Kind

The Meixner polynomials of the second kind are defined bythe following recurrence relation [23 24]

119872119899+1

(119909 120575 120578) = (119909 minus 119887119899)119872119899(119909 120575 120578) minus 120582

119899119872119899minus1

(119909 120575 120578)

(23)

where1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 119887

0

119887119899= (2119899 + 120578) 120575

120582119899= (1205752

+ 1) 119899 (119899 + 120578 minus 1)

(24)

Using (23) we get the first few Meixner polynomials ofthe second kind

1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 120575120578

1198722(119909 120575 120578) = 119909

2

minus 1199092120575 (1 + 120578) + 120578 ((120578 + 1) 1205752

minus 1)

(25)

The Meixner polynomials of the second kind are definedby the following generating function [22]

infin

sum119899=0

119872119899(119909 120575 120578)

119899119905119899

= ((1 + 120575119905)2

+ 1199052

)minus1205782

sdot exp(119909tanminus1 ( 119905

1 + 120575119905))

(26)



We represent the generating function (26) as a product ofgenerating functions 119860(119905) 119861(119905) where

119860 (119905) = ((1 + 120575119905)2

+ 1199052

)minus1205782

= (1 + 2120575119905 + (1205752

+ 1) 1199052

)minus1205782

119861 (119905) = exp (119909tanminus1 ( 119905

1 + 120575119905))

(27)

Next we represent 119860(119905) as a composition of generatingfunctions 119860

1(1198602(119905)) and we expand 119860

1(119905) by binomial

theorem

1198601(119905) = (1 + 119905)

minus1205782

=

infin

sum119899=0

(minus120578

2119899

) 119905119899

1198602(119905) = 2120575119905 + (120575

2

+ 1) 1199052

(28)

Coefficients of generating function 1198601(119905) = sum

infin

119899=01198861(119899)119905119899

are

1198861(119899) = (

minus120578

2119899

) (29)

The composita for the generating function 119886119909 + 1198871199092 isgiven as follows (see [15])

1198862119896minus119899

119887119899minus119896

(119896

119899 minus 119896) (30)

Then composita of the generating function 1198602(119905) equals

119860Δ

2(119899 119896) = (2120575)

2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896) (31)

Using (5) we obtain coefficients of the generating func-tion 119860(119905) = sum

infin

119899=0119886(119899)119905119899

119886 (0) = 1198861(0)

119886 (119899) =

119899

sum119896=1

119860Δ

2(119899 119896) 119886

1(119896)

(32)

Therefore we get the following expression

119886 (0) = 1

119886 (119899) =

119899

sum119896=1

(2120575)2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896)(

minus120578

2119896

) (33)


1(1198612(119905)) where

1198611(119905) = 119890

119905

=

infin

sum119899=0

1

119899119905119899

1198612(119905) = 119909tanminus1 ( 119905

1 + 120575119905)

(34)


We also represent 1198612(119905) as a composition of generating

functions 1199091198621(1198622(119905)) where

1198621(119905) = tanminus1 (119905)

1198622(119905) =

119905

1 + 120575119905

(35)

The composita for the generating function tanminus1(119905) isgiven as follows (see [19])

((minus1)(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896] (36)

where [119895

119896] is the Stirling number of the first kind

Then composita of generating function 1198621(119905) equals

119862Δ

1(119899 119896) = ((minus1)

(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

sdot

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896]

(37)

The composita for the generating function 119886119905(1 minus 119887119905) isgiven as follows (see [19])

(119899 minus 1

119896 minus 1) 119886119896

119887119899minus119896

(38)

Therefore composita of generating function1198622(119905) is equal

to

119862Δ

2(119899 119896) = (

119899 minus 1

119896 minus 1) (minus120575)

119899minus119896

(39)

Using rules (4) and (2) we obtain composita of generatingfunction 119861

2(119905)

119861Δ

2(119899 119896) = 119909

119896

119899

sum119898=119896

119862Δ

2(119899119898) 119862

Δ

1(119898 119896) (40)


infin

119899=01198871(119899)119905119899

are defined by

1198871(119899) =

1

119899 (41)

Then using rule (5) we obtain coefficients of generatingfunction 119861(119905) = sum

infin

119899=0119887(119899)119905119899

119887 (0) = 1198871(0)

119887 (119899) =

119899

sum119896=1

119861Δ

2(119899 119896) 119887

1(119896)

(42)

After some transformations we obtain the followingexpression

119887 (0) = 1

119887 (119899) =

119899

sum119896=1

119909119896

119899

119899

sum119898=119896

(119899

119898)120575119899minus119898

(1 + (minus1)

119898+119896

(minus1)(119898+119896)2minus119899

)

sdot

119898

sum119895=119896


(119895 minus 1)(119898

119895)[

119895

119896]

(43)

Therefore using (6) we obtain a new explicit formula forthe Meixner polynomials of the second kind

119872119899(119909 120575 120578) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896) (44)



Acknowledgment

This work was partially supported by the Ministry of Edu-cation and Science of Russia Government Order no 3657(TUSUR)

References

[1] R P Boas Jr and R C Buck Polynomial Expansions of AnalyticFunctions Springer 1964

[2] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited ChichesterUK John Wiley amp Sons New York NY USA 1984

[3] H M Srivastava and J Choi Zeta and Q-Zeta Functions andAssociated Series and Integrals Elsevier Science PublishersAmsterdam The Netherlands 2012

[4] H M Srivastava ldquoSome generalizations and basic (or q-)extensions of the Bernoulli Euler and Genocchi polynomialsrdquoApplied Mathematics and Information Sciences vol 5 pp 390ndash444 2011

[5] Y Simsek and M Acikgoz ldquoA new generating function of (q-)Bernstein-type polynomials and their interpolation functionrdquoAbstract and Applied Analysis vol 2010 Article ID 769095 12pages 2010

[6] H Ozden Y Simsek and H M Srivastava ldquoA unified presen-tation of the generating functions of the generalized BernoulliEuler and Genocchi polynomialsrdquo Computers amp Mathematicswith Applications vol 60 no 10 pp 2779ndash2787 2010

[7] R Dere and Y Simsek ldquoApplications of umbral algebra tosome special polynomialsrdquo Advanced Studies in ContemporaryMathematics vol 22 no 3 pp 433ndash438 2012

[8] Y Simsek ldquoComplete sum of products of (hq)-extension ofEuler polynomials and numbersrdquo Journal of Difference Equa-tions and Applications vol 16 no 11 pp 1331ndash1348 2010

[9] Y He and S Araci ldquoSums of products of APOstol-Bernoulli andAPOstol-Euler polynomialsrdquo Advances in Difference Equationsvol 2014 article 155 2014

[10] S Araci ldquoNovel identities involving Genocchi numbers andpolynomials arising from applications of umbral calculusrdquoApplied Mathematics and Computation vol 233 pp 599ndash6072014

[11] H M Srivastava and G-D Liu ldquoExplicit formulas for theNorlund polynomials 119861(119909)

119899and 119887(119909)119899rdquo Computers amp Mathematics

with Applications vol 51 pp 1377ndash1384 2006[12] H M Srivastava and P G Todorov ldquoAn explicit formula for the

generalized Bernoulli polynomialsrdquo Journal of MathematicalAnalysis and Applications vol 130 no 2 pp 509ndash513 1988

[13] M Cenkci ldquoAn explicit formula for generalized potentialpolynomials and its applicationsrdquo Discrete Mathematics vol309 no 6 pp 1498ndash1510 2009


[14] K N Boyadzhiev ldquoDerivative polynomials for tanh tan sechand sec in explicit formrdquoThe Fibonacci Quarterly vol 45 no 4pp 291ndash303 (2008) 2007

[15] D V Kruchinin and V V Kruchinin ldquoApplication of a com-position of generating functions for obtaining explicit formulasof polynomialsrdquo Journal of Mathematical Analysis and Applica-tions vol 404 no 1 pp 161ndash171 2013

[16] D V Kruchinin and V V Kruchinin ldquoExplicit formulas forsome generalized polynomialsrdquo Applied Mathematics amp Infor-mation Sciences vol 7 no 5 pp 2083ndash2088 2013

[17] D V Kruchinin ldquoExplicit formula for generalizedMott polyno-mialsrdquoAdvanced Studies in Contemporary Mathematics vol 24no 3 pp 327ndash322 2014

[18] D V Kruchinin and V V Kruchinin ldquoA method for obtain-ing expressions for polynomials based on a composition ofgenerating functionsrdquo in Proceedings of the AIP Conference onNumerical Analysis and Applied Mathematics (ICNAAM rsquo12)vol 1479 pp 383ndash386 2012

[19] V V Kruchinin and D V Kruchinin ldquoComposita and itspropertiesrdquo Journal of Analysis and Number Theory vol 2 no2 pp 37ndash44 2014

[20] H L Krall and O Frink ldquoA new class of orthogonal polyno-mials the Bessel polynomialsrdquo Transactions of the AmericanMathematical Society vol 65 pp 100ndash115 1949

[21] L Carlitz ldquoA note on the Bessel polynomialsrdquo Duke Mathemat-ical Journal vol 24 pp 151ndash162 1957

[22] S RomanThe Umbral Calculus Academic Press 1984[23] G Hetyei ldquoMeixner polynomials of the second kind and

quantum algebras representing su(11)rdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol466 no 2117 pp 1409ndash1428 2010

[24] T S Chihara An Introduction to Orthogonal PolynomialsGordon and Breach Science 1978

Research ArticleFaber Polynomial Coefficients of Classes ofMeromorphic Bistarlike Functions

Jay M Jahangiri1 and Samaneh G Hamidi2

1Department of Mathematical Sciences Kent State University Burton OH 44021-9500 USA2Department of Mathematics Brigham Young University Provo UT 84602 USA

Correspondence should be addressed to Jay M Jahangiri jjahangikentedu

Received 15 December 2014 Accepted 8 January 2015

Academic Editor A Zayed

Copyright copy 2015 J M Jahangiri and S G Hamidi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Applying the Faber polynomial coefficient expansions to certain classes of meromorphic bistarlike functions we demonstrate theunpredictability of their early coefficients and also obtain general coefficient estimates for such functions subject to a given gapseries condition Our results improve some of the coefficient bounds published earlier

Let Σ be the family of functions 119892 of the form

119892 (119911) =1

119911+ 1198870+

infin

sum119899=1

119887119899119911119899 (1)

that are univalent in the punctured unit disk D = 119911 0 lt

|119911| lt 1For the real constants 119860 and 119861 (0 le 119861 le 1 minus119861 le 119860 lt 119861)

let Σ(119860 119861) consist of functions 119892 isin Σ so that

1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911) (2)

where 120593(119911) = suminfin

119899=1119888119899119911119899 is a Schwarz function that is 120593(119911) is

analytic in the open unit disk |119911| lt 1 and |120593(119911)| le |119911| lt 1Note that |119888

119899| le 1 (Duren [1]) and the functions in Σ(119860 119861)

are meromorphic starlike in the punctured unit disk D (egsee Clunie [2] and Karunakaran [3]) It has been proved byLibera and Livingston [4] and Karunakaran [3] that |119887

119899| le

(119861 minus 119860)(119899 + 1) for 119892 isin Σ[119860 119861]

The coefficients of ℎ = 119892minus1 the inverse map of 119892 are

given by the Faber polynomial expansion (eg see Airaultand Bouali [5] or Airault and Ren [6 page 349])

ℎ (119908) = 119892minus1

(119908) =1

119908+

infin

sum119899=0

119861119899119908119899

=1

119908minus 1198870minus sum119899ge1

1

119899119870119899

119899+1(1198870 1198871 119887

119899) 119908119899

(3)

where 119908 isin D

119870119899

119899+1(1198870 1198871 119887

119899)

= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)

+119899 (119899 minus 1) (119899 minus 2) (119899 minus 3)

3119887119899minus4

0(1198874+ 311988711198872)

+ sum119895ge5

119887119899minus119895

0119881119895

(4)

and 119881119895is a homogeneous polynomial of degree 119895 in the

variables 1198871 1198872 119887

119899



In 1923 Lowner [7] proved that the inverse of the Koebefunction 119896(119911) = 119911(1minus119911)

2 provides the best upper bounds forthe coefficients of the inverses of analytic univalent functionsAlthough the estimates for the coefficients of the inversesof analytic univalent functions have been obtained in asurprisingly straightforward way (eg see [8 page 104])the case turns out to be a challenge when the biunivalencycondition is imposed on these functions A function is saidto be biunivalent in a given domain if both the functionand its inverse are univalent there By the same token afunction is said to be bistarlike in a given domain if both thefunction and its inverse are starlike there Finding boundsfor the coefficients of classes of biunivalent functions datesback to 1967 (see Lewin [9]) The interest on the boundsfor the coefficients of subclasses of biunivalent functionspicked up by the publications [10ndash14] where the estimatesfor the first two coefficients of certain classes of biunivalentfunctions were provided Not much is known about thehigher coefficients of the subclasses biunivalent functions asAli et al [13] also declared that finding the bounds for |119886

119899|

119899 ge 4 is an open problem In this paper we use the Faberpolynomial expansions of the functions 119892 and ℎ = 119892

minus1 inΣ(119860 119861) to obtain bounds for their general coefficients |119886

119899| and

provide estimates for the early coefficients of these types offunctions

We will need the following well-known two lemmas thefirst of which can be found in [15] (also see Duren [1])

Lemma 1 Let 119901(119911) = 1 + suminfin

119899=1119901119899119911119899 be so that 119877119890(119901(119911)) gt 0

for |119911| lt 1 If 120572 ge minus12 then

100381610038161003816100381610038161199012+ 1205721199012

1

10038161003816100381610038161003816le 2 + 120572

1003816100381610038161003816119901110038161003816100381610038162

(5)

Consequently we have the following lemma in which wewill provide a short proof for the sake of completeness

Lemma 2 Consider the Schwarz function 120593(119911) = suminfin

119899=1119888119899119911119899

where |120593(119911)| lt 1 for |119911| lt 1 If 120574 ge 0 then

100381610038161003816100381610038161198882+ 1205741198882

1

10038161003816100381610038161003816le 1 + (120574 minus 1)

1003816100381610038161003816119888110038161003816100381610038162

(6)

Proof Write

119901 (119911) =[1 + 120593 (119911)]

[1 minus 120593 (119911)] (7)

where 119901(119911) = 1+suminfin

119899=1119901119899119911119899 is so that Re(119901(119911)) gt 0 for |119911| lt 1

Comparing the corresponding coefficients of powers of 119911 in119901(119911) = [1 + 120593(119911)][1 minus 120593(119911)] shows that 119901

1= 21198881and 119901

2=

2(1198882+ 11988821)

By substituting for 1199011= 21198881and 119901

2= 2(1198882+ 11988821) in (5) we

obtain100381610038161003816100381610038162 (1198882+ 1198882

1) + 120572 (2119888

1)210038161003816100381610038161003816le 2 + 120572

10038161003816100381610038162119888110038161003816100381610038162 (8)

or100381610038161003816100381610038161198882+ (1 + 2120572) 119888

2

1

10038161003816100381610038161003816le 1 + 2120572

1003816100381610038161003816119888110038161003816100381610038162

(9)

Now (6) follows upon substitution of 120574 = 1 + 2120572 ge 0 in theabove inequality

In the following theorem we will observe the unpre-dictability of the early coefficients of the functions 119892 and itsinverse map ℎ = 119892minus1 in Σ[119860 119861] providing an estimate for thegeneral coefficients of such functions

Theorem 3 For 0 le 119861 le 1 and minus119861 le 119860 lt 119861 let the function119892 and its inverse map ℎ = 119892minus1 be in Σ[119860 119861] Then

(119894)10038161003816100381610038161198870

1003816100381610038161003816 le

119861 minus 119860

radic2119861 minus 119860 119894119891 2119861 minus 119860 ge 1

119861 minus 119860 119900119905ℎ119890119903119908119894119904119890

(119894119894)10038161003816100381610038161198871

1003816100381610038161003816

le

119861 minus 119860

2minus1 minus 119860 minus 2119861

2 (119861 minus 119860)|1198870|2

119894119891 0 le 119860 le 1 minus 2119861

119861 minus 119860

2 119900119905ℎ119890119903119908119894119904119890

(119894119894119894)100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2

(119894V) 10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 119894119891 119887

119896= 0 119891119900119903 0 le 119896 le 119899 minus 1

(10)

Proof Consider the function 119892 isin Σ given by (1) Therefore(see [5 6])

1199111198921015840 (119911)

119892 (119911)= minus1 minus

infin

sum119899=0

119865119899+1

(1198870 1198871 1198872 119887

119899) 119911119899+1

(11)

where 119865119899+1

(1198870 1198871 1198872 119887

119899) is a Faber polynomial of degree

119899 + 1 We note that 1198651= minus1198870 1198652= 11988720minus 21198871 1198653= minus11988730+

311988711198870minus 31198872 1198654= 11988740minus 4119887201198871+ 411988701198872+ 211988721minus 41198873 and 119865

5=

minus11988750+ 5119887301198871minus 5119887201198872minus 5119887011988721+ 511988711198872+ 511988701198873minus 51198874 In general

(Bouali [16 page 52])

119865119899+1

(1198870 1198871 119887

119899)

= sum1198941+21198942+sdotsdotsdot+(119899+1)119894

119899+1=119899+1

119860 (1198941 1198942 119894

119899+1) 1198871198941

01198871198942

1sdot sdot sdot 119887119894119899+1

119899

(12)

where

119860 (1198941 1198942 119894

119899+1)

= (minus1)(119899+1)+2119894

1+sdotsdotsdot+(119899+2)119894

119899+1

(1198941+ 1198942+ sdot sdot sdot + 119894

119899+1minus 1) (119899 + 1)

(1198941) (1198942) sdot sdot sdot (119894

119899+1)

(13)


Similarly for the inverse map ℎ = 119892minus1 we have

119908ℎ1015840 (119908)

ℎ (119908)= minus1 minus

infin

sum119899=1

119865119899+1

(1198610 1198611 1198612 119861

119899) 119908119899+1

(14)

where119865119899+1

(1198610 1198611 1198612 119861


119899 + 1 given by

119865119899+1

= minus119899 (119899 minus (119899 + 1))

119899 (119899 minus 2119899)119861119899

0

minus119899 (119899 minus (119899 + 1))

(119899 minus 2) (119899 minus (2119899 minus 1))119861119899minus2

01198611

minus119899 (119899 minus (119899 + 1))


01198612

minus119899 (119899 minus (119899 + 1))


0

sdot (1198613+119899 minus (2119899 minus 3)

21198612

1) minus sum119895ge5

119861119899minus119895

0119870119895

(15)

where 119870119895is a homogeneous polynomial of degree 119895 in the

variables 1198611 1198612 119861

119899minus1and

1198610= minus1198870 119861

119899= minus

1

119899119870119899

119899+1(1198870 1198871 119887

119899) (16)

Since both 119892 and its inverse map ℎ = 119892minus1 are in Σ[119860 119861] theFaber polynomial expansion yields (also see Duren [1 pages118-119])

1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198881 1198882 119888

119899 119861) 119911119899

(17)

119908ℎ1015840 (119908)

ℎ (119908)= minus

1 + 119860120595 (119908)

1 + 119861120595 (119908)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198891 1198892 119889

119899 119861) 119908

119899

(18)


119899=1119888119899119911119899 and 120595(119908) = sum

infin

119899=1119889119899119908119899 are two

Schwarz functions that is |120593(119911)| lt 1 for |119911| lt 1 and |120595(119908)| lt1 for |119908| lt 1

In general (see Airault [17] or Airault and Bouali [5]) thecoefficients 119870119901

119899(1198961 1198962 119896

119899 119861) are given by

119870119901

119899(1198961 1198962 119896

119899 119861)

=119901

(119901 minus 119899)119899119896119899

1119861119899minus1

+119901

(119901 minus 119899 + 1) (119899 minus 2)119896119899minus2

11198962119861119899minus2

+119901


11198963119861119899minus3

+119901


1

sdot [1198964119861119899minus4

+119901 minus 119899 + 3

21198962

2119861]

+119901


1

sdot [1198965119861119899minus5

+ (119901 minus 119899 + 4) 11989621198963119861]

+ sum119895ge6

119896119899minus119895

1119883119895

(19)


variables 1198962 1198963 119896

119899

Comparing the corresponding coefficients of (11) and (17)implies

minus119865119899+1

(1198870 1198871 1198872 119887

119899) = (119860 minus 119861)119870

minus1

119899+1(1198881 1198882 119888

119899+1 119861)

(20)

Similarly comparing the corresponding coefficients of(14) and (18) gives

minus 119865119899+1

(1198610 1198611 1198612 119861

119899)

= (119860 minus 119861)119870minus1

119899+1(1198891 1198892 119889

119899+1 119861)

(21)

Substituting 119899 = 0 119899 = 1 and 119899 = 2 in (16) (20) and (21)respectively yields

1198870= (119860 minus 119861) 119888

1

minus1198870= (119860 minus 119861) 119889

1

(22)

21198871minus 1198872

0= (119861 minus 119860) (119888

2

1119861 minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (119889

2

1119861 minus 1198892)

(23)

Taking the absolute values of either equation in (22) weobtain |119887

0| le 119861 minus 119860 Obviously from (22) we note that

1198881= minus1198891 Solving the equations in (23) for 1198872

0and then adding

them gives

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus 1198611198882

1minus 1198611198892

1) (24)


Now in light of (22) we conclude that

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus

2119861

(119861 minus 119860)21198872

0) (25)

Once again solving for 11988720and taking the square root of both

sides we obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(119861 minus 119860)

2

(1198882+ 1198892)

2 (2119861 minus 119860)le

119861 minus 119860

radic2119861 minus 119860 (26)

Now the first part of Theorem 3 follows since for 2119861 minus 119860 gt 1

it is easy to see that

119861 minus 119860

radic2119861 minus 119860lt 119861 minus 119860 (27)

Adding the equations in (23) and using the fact that 1198881= minus1198891

we obtain

41198871= (119861 minus 119860) [(119889

2minus 1198882) minus (119889

2

1minus 1198882

1) 119861] = (119861 minus 119860) (119889

2minus 1198882)

(28)

Dividing by 4 and taking the absolute values of both sidesyield

100381610038161003816100381611988711003816100381610038161003816 le

119861 minus 119860

2 (29)

On the other hand from the second equations in (22) and(23) we obtain

21198871= (119861 minus 119860) (119889

2+ 1198601198892

1minus 2119861119889

2

1) (30)

Taking the absolute values of both sides and applyingLemma 2 it follows that

100381610038161003816100381611988711003816100381610038161003816 le

1

2(119861 minus 119860) (

100381610038161003816100381610038161198892+ 1198601198892

1

10038161003816100381610038161003816+ 2119861

1003816100381610038161003816119889110038161003816100381610038162

)

le1

2(119861 minus 119860) (1 + (119860 minus 1)

1003816100381610038161003816119889110038161003816100381610038162

+ 211986110038161003816100381610038161198891

10038161003816100381610038162

)

=119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

(31)

This concludes the second part of Theorem 3 since for 0 lt

119860 lt 1 minus 2119861 we have119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

lt119861 minus 119860

2 (32)

Substituting (22) in (23) we obtain

21198871minus 1198872

0= (119861 minus 119860) (

119861

(119861 minus 119860)21198872

0minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (

119861

(119860 minus 119861)21198872

0minus 1198892)

(33)

Following a simple algebraic manipulation we obtain thecoefficient body

100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2 (34)

Finally for 119887119896= 0 0 le 119896 le 119899 minus 1 (20) yields

(119899 + 1) 119887119899= (119860 minus 119861) 119888

119899+1 (35)

Solving for 119887119899and taking the absolute values of both sides we

obtain

10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 (36)

Remark 4 The estimate |1198870| le (119861 minus 119860)radic2119861 minus 119860 given by

Theorem 3(i) is better than that given in ([14Theorem 2(i)])

Remark 5 In ([3Theorem 1]) the bound |119887119899| le (119861minus119860)(119899+1)

was declared to be sharp for the coefficients of the function119892 isin Σ[119860 119861]The coefficient estimates |119887

0| le (119861minus119860)radic2119861 minus 119860

and |1198871| le (119861 minus 119860)2 minus ((1 minus 119860 minus 2119861)2(119861 minus 119860))|119887

0|2 given by

Theorem 3 show that the coefficient bound |119887119899| le (119861minus119860)(119899+

1) is not sharp for the meromorphic bistarlike functions thatis if both 119892 and its inverse map 119892

minus1 are in Σ[119860 119861] Findingsharp coefficient bound formeromorphic bistarlike functionsremains an open problem



References

[1] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[2] J Clunie ldquoOn meromorphic schlicht functionsrdquo Journal of theLondon Mathematical Society vol 34 pp 215ndash216 1959

[3] V Karunakaran ldquoOn a class of meromorphic starlike functionsin the unit discrdquoMathematical Chronicle vol 4 no 2-3 pp 112ndash121 1976

[4] R J Libera and A E Livingston ldquoBounded functions with pos-itive real partrdquo Czechoslovak Mathematical Journal vol 22 pp195ndash209 1972

[5] H Airault and A Bouali ldquoDifferential calculus on the Faberpolynomialsrdquo Bulletin des Sciences Mathematiques vol 130 no3 pp 179ndash222 2006

[6] H Airault and J Ren ldquoAn algebra of differential operators andgenerating functions on the set of univalent functionsrdquo Bulletindes Sciences Mathematiques vol 126 no 5 pp 343ndash367 2002

[7] K Lowner ldquoUntersuchungen uber schlichte konforme Abbil-dungen des Einheitskreises IrdquoMathematische Annalen vol 89no 1-2 pp 103ndash121 1923

[8] J G Krzyz R J Libera and E Złotkiewicz ldquoCoefficients ofinverses of regular starlike functionsrdquo Annales UniversitatisMariae Curie-Skłodowska A vol 33 pp 103ndash110 1979

[9] M Lewin ldquoOn a coefficient problem for bi-univalent functionsrdquoProceedings of the American Mathematical Society vol 18 no 1pp 63ndash63 1967

[10] D A Brannan and T S Taha ldquoOn some classes of bi-univalent functionsrdquo Studia Universitatis Babes-Bolyai SeriesMathematica vol 31 no 2 pp 70ndash77 1986


[11] HM Srivastava A KMishra and P Gochhayat ldquoCertain sub-classes of analytic and bi-univalent functionsrdquo Applied Math-ematics Letters vol 23 no 10 pp 1188ndash1192 2010

[12] B A Frasin and M K Aouf ldquoNew subclasses of bi-univalentfunctionsrdquoAppliedMathematics Letters vol 24 no 9 pp 1569ndash1573 2011

[13] R M Ali S K Lee V Ravichandran and S SupramaniamldquoCoefficient estimates for bi-univalent Ma-Minda starlike andconvex functionsrdquo Applied Mathematics Letters vol 25 no 3pp 344ndash351 2012

[14] S G Hamidi S A Halim and J M Jahangiri ldquoFaber poly-nomial coefficient estimates for meromorphic bi-starlike func-tionsrdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 498159 4 pages 2013

[15] M Jahangiri ldquoOn the coefficients of powers of a class of Bazile-vic functionsrdquo Indian Journal of Pure and Applied Mathematicsvol 17 no 9 pp 1140ndash1144 1986

[16] A Bouali ldquoFaber polynomials Cayley-Hamilton equationand Newton symmetric functionsrdquo Bulletin des SciencesMathematiques vol 130 no 1 pp 49ndash70 2006

[17] H Airault ldquoRemarks on Faber polynomialsrdquo InternationalMathematical Forum vol 3 no 9ndash12 pp 449ndash456 2008

Research ArticleSome Properties of Certain Class of Analytic Functions

Uzoamaka A Ezeafulukwe12 and Maslina Darus2

1 Mathematics Department Faculty of Physical Sciences University of Nigeria Nsukka Nigeria2 School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia43600 Bangi Selangor Malaysia

Correspondence should be addressed to Maslina Darus maslinaukmedumy

Received 21 April 2014 Revised 18 June 2014 Accepted 27 June 2014 Published 10 July 2014


Copyright copy 2014 U A Ezeafulukwe and M Darus This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We obtain some properties related to the coefficient bounds for certain subclass of analytic functions We also work on thedifferential subordination for a certain class of functions

1 Introductions

Let 119867 denote the class of functions

119891 (119911) = 119911 +

infin

sum120581=2

119886120581119911120581

(1)

which is analytic in the unit disc 119880 = 119911 isin C |119911| lt 1 Let

119867[119886 119899] = 119901 isin 119867 119901 (119911) = 119886 + 119886119899119911119899

+119886119899+1

119911119899+1

+ sdot sdot sdot 119899 isin N 119886 = 0

(2)

Now let 119867(120573) be the class of functions defined by

119865 (119911) = 119911120573

+

infin

sum120581=2

120582120581119886120581119911120573+120581minus1

120573 isin N

120582120581= 120573120581minus1

119911 =1

120573 120573 |119911| lt 1

(3)

The Hadamard product 119891 lowast 119892 of two functions 119891 and 119892 isdefined by

(119891 lowast 119892) (119911) =

infin

sum119896=0

119886119896119887119896119911119896

(4)


119896=0119886119896119911119896 and 119892(119911) = sum

infin

119896=0119887119896119911119896 are analytic in

119880

Let 120595(119911) = 119911120573 + sum

infin

120581=2119886120581119911120573+120581minus1 120573 isin N and then 120595(119911) is

analytic in the open unit disc119880 The function 119865(119911) defined in(3) is equivalent to

120595 (119911) lowast119911120573

1 minus 120573119911 119911 =

1

120573 120573 isin N 120573 |119911| lt 1 (5)

where lowast is the Hadamard product and 119865(119911) is analytic in theopen unit disc 119880

We introduce a class of functions

119876(120485) (120572 120591 120574 120573)

= 119865 (119911) isin 119867 (120573) R119890

times120572[119865(119911)]

(120485)

[119911120573](120485)

+120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

gt 120574 119911 isin 119880

(6)

where

119911 = 0 120573 gt 120485 120485 isin N cup 0 also 120572 + 120591 = 120574 (7)

Authors like Saitoh [1] and Owa [2 3] had previously stud-ied the properties of the class of functions 119876(0)(120572 120591 120574 1)They obtained many interesting results and Wang et al [4]



studied the extreme points coefficient bounds and radius ofunivalency of the same class of functions They obtained thefollowing theorem among other results

Theorem 1 (see [4]) Let 119891(119911) isin 119867 A function 119891(119911) isin

119876(0)(120572 120591 120574 1) if and only if 119891(119911) can be expressed as

119891 (119911) =1

120572 + 120591int|119909|

[ (2120574 minus 120572 minus 120591) 119911 + 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581

119911120581+1

(120581 + 1) 120591 + 120572] 119889120583 (119909)

(8)

where 120583(119909) is the probability measure defined on120594 = 119909 |119909| = 1 (9)

For fixed120572 120591 and 120574 the class119876(0)(120572 120591 120574 1) and the probabilitymeasure 120583 defined on 120594 are one-to-one by expression (8)

Recently Hayami et al [5] studied the coefficient esti-mates of the class of function 119891(119911) isin 119867 in the open unitdisc 119880 They derived results based on properties of the classof functions 119891(119911) isin 119867[119886 119899] 119886 = 0 Xu et al [6] used the prin-ciple of differential subordination and the Dziok-Srivastavaconvolution operator to investigate some analytic propert-ies of certain subclass of analytic functions We also note thatStanciu et al [7] used the properties of the class of functions119891(119911) isin 119867[119886 119899] 119886 = 0 to investigate the analytic and univalentproperties of the following integral operator

Θ120572120573

(119911) = (120573int119911

0

119905120573minus120572minus1

[119891 (119905)]120572

119892 (119905) 119889119905)

1120573

(10)

where (120572 isin C 120573 isin C 0 119891 isin 119867 119892 isin 119867[1 119899])Motivated by the work in [1ndash7] we used the properties of

the class of function 119891(119911) isin 119867[119886 119899] 119886 = 0 to investigate thecoefficient estimates of the class of functions 119865(119911) isin 119867(120573) inthe open unit disc 119880 We also use the principle of differentialsubordination to investigate some properties of the class offunctions 119876(120485)(120572 120591 120574 120573)

We state the following known results required to proveour work

Definition 2 If 119892 and ℎ are analytic in 119880 then 119892 is said tobe subordinate to ℎ written as 119892 ≺ ℎ or 119892(119911) ≺ ℎ(119911) If 119892 isunivalent in 119880 then 119892(0) = ℎ(0) and 119892(119880) sub ℎ(119880)

Theorem 3 (see [8]) Consider 119901 isin 119867[1 119899] if and only if thereis probability measure 120583 on 120594 such that

119901 (119911) = int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909) (|119911| lt 1) (11)

and120594 = 119909 |119909| = 1The correspondence between119867[1 119899] andthe set of probability measures 120583 on 120594 given by Hallenbeck [9]is one-to-one

Theorem 4 (see [10 11]) Let ℎ(119911) be convex in 119880 ℎ(0) = 119886119888 = 0 andR119890119888 ge 0 If 119892(119911) isin 119867[119886 119899] and

119892 (119911) +1199111198921015840 (119911)

119888≺ ℎ (119911) (12)

then

119892 (119911) ≺ 119902 (119911) =119888119911minus119888120581

120581int119911

0

119905(119888minus120581)120581

ℎ (119905) 119889119905 ≺ ℎ (119911) (13)

The function 119902 is convex and the best (119886 119899)-dominant

Lemma 5 (see [10]) Let ℎ be starlike in 119880 with ℎ(0) = 0 and119886 = 0 If 119901 isin 119867[119886 119899] satisfies

1199111199011015840 (119911)

119901 (119911)≺ ℎ (119911) (14)

then

119901 (119911) ≺ 119902 (119911) = 119886 exp [1

119899int119911

0

ℎ (120585) 120585minus1

119889120585] (15)

and 119902 is the best (119886 119899)-dominant

Lemma 6 (see [12]) Let 119901 isin [1 119899] with R119890119901(119911) gt 0 in 119880Then for |119911| = 119903 lt 1

(i) (1 minus 119903)(1 + 119903) le R119890119901(119911) le |119901(119911)| le (1 + 119903)(1 minus 119903)

(ii) |1199011015840(119911)| le 2R119890119901(119911)(1 minus 1199032)

Remark 7 The combination (i) and (ii) of Lemma 6 gives

100381610038161003816100381610038161003816100381610038161003816

1199111199011015840 (119911)

119901 (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 119901 (119911)

(1 minus 119903)2

(16)

Remark 8 For convenience we limit our result to the prin-cipal branch and otherwise stated the constrains on 120573 120485 120572 120591120574 and 120582

120581which remain the same throughout this paper

2 Coefficient Bounds of the Class ofFunctions 119876

(120485)

(120572120591120574120573)

We begin with the following result

Theorem 9 Let 119865(119911) be as defined in (3) A function 119865(119911) isin

119876(120485)(120572 120591 120574 120573) if and only if 119911120485[119865(119911)](120485) can be expressed as

119911120485

[119865 (119911)](120485)

=

120573

119875120485

(120572 + 120591)int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120573 minus 120485) (120572 + 120591) 119909120581119911120573+120581

(120572 + 120591) (120573 minus 120485) + 120581120591] 119889120583 (119909)

(17)

where 119899119875119903

= 119899(119899 minus 119903) and 120583 is the probability measuredefined on 120594 = 119909 |119909| = 1


Proof If 119865(119911) isin 119876(120485)(120572 120591 120574 120573) then

119901 (119911)

=

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

isin 119867 [1 119899]

(18)

ByTheorem 3

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

= int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909)

(19)

and (19) can be written as

[119865 (119911)](120485+1)

[119911120573](120485+1)

+120572[119865 (119911)]

(120485)

120591[119911120573](120485)

=1

120591int|119909|=1

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909119911

1 minus 119909119911119889120583 (119909)

(20)

which yields

119911((120572+120591)(120485minus120573)+120591)120591

int119911

0

[[119865 (120585)]

(120485+1)

[120585120573](120485+1)

+120572[119865 (120585)]

(120485)

120591[120585120573](120485)

]

times 120585((120572+120591)(120573minus120485)minus120591)120591

119889120585

=1

120591int|119909|=1

119911((120572+120591)(120485minus120573)+120591)120591

times int119911

0

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909120585

1 minus 119909120585


119889120585 119889120583 (119909)

(21)

and so the expression (17)

If 119911120485[119865(119911)](120485) can be expressed as (17) reverse calculation

shows that 119865(119911) isin 119876(120485)(120572 120591 120574 120573)

Corollary 10 Let 119865 be defined as in (3) A function 119865(119911) isin


119865 (119911) =1

120572 + 120591int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581119911120573+120581

120572120573 + 120591 (120581 + 120573)] 119889120583 (119909)

(22)

where 120583 is the probability measure defined on 120594 = 119909 |119909| =

1

Proof It is as in Theorem 9

Corollary 11 Let 119865(119911) be as defined in (3) If 119865(119911) isin 119876(120485)(120572 120591

120574 120573) then for 120581 ge 2 and 119899119875119903= 119899(119899 minus 119903) we have

10038161003816100381610038161198861205811003816100381610038161003816 le 119867 (120572 120591 120574 120573) (23)

where

119867(120572 120591 120574 120573) =2 (120572 + 120591 minus 120574) (120573 minus 120485) [

120573

119875120485]

120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(24)

Proof Let 119865(119911) isin 119876(120485)(120572 120591 120574 120573) from (17) and

119911120485

[119865119909(119911)](120485)

120573119875120485

= 119911120573

+

infin

sum120581=2

2 (120572 + 120591 minus 120574) (120573 minus 120485) 119909120581minus1119911120573+120581minus1

(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)(|119909| = 1)

(25)

Comparing the coefficient yields the result

Theorem12 Let119866(119911) isin 119876(120485)(120572 120591 120574 120573) andΨ(119911) = 119866(119911)(120572+

120591 minus 120574) Then for |119911| = 119903 lt 1 we have100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [[120581 minus 1] 119903

120581minus1]

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

le2119903R119890 Ψ (119911)

(1 minus 119903)2

(26)

Proof Since 119866(119911) isin 119876(120485)(120572 120591 120574 120573) then

119866 (119911) equiv120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

+120572[119865(119911)]

(120485)

[119911120573](120485)

minus 120574 (27)

and then119911Ψ1015840 (119911)

Ψ (119911)=

suminfin

120581=2120599 (120572 120591 120574 120573) [120581 minus 1] 119886

120581119911120581minus1

1 + suminfin

120581=2120599 (120572 120591 120574 120573) 119886

120581119911120581minus1

(28)

where

120599 (120572 120591 120574 120573) =120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591 minus 120574) [120573

119875(120485)

]

(29)

From (23) and (28) we got100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [120581 minus 1] 119903

120581minus1

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

(120581 ge 2) (30)

The application of Remark 7 to (27) gives100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 Ψ (119911)

(1 minus 119903)2

(31)

Sinceinfin

sum120581=2

2 (120573 minus 120485) [120581 minus 1] 119903120581minus1

lt 2119903R119890 Ψ (119911)

1 +

infin

sum120581=2

2 (120573 minus 120485) 119903120581minus1

gt (1 minus 119903)2

(32)

thenTheorem 12 is proved


3 Application of Differential Subordination tothe Function 119876

(120485)

(120572 120591 120574 120573)

Here we calculate some subordinate properties of the class119876(120485)(120572 120591 120574 120573)

Theorem 13 Let119866(119911) isin 119876(120485)(120572 120591 120574 120573) and let ℎ(119911) be starlikein 119880 with ℎ(0) = 0 and 120572 + 120591 = 120574 If

1199111198661015840 (119911)

119866 (119911) ≺ ℎ (119911) (33)

then

119866 (119911) ≺ 119902 (119911) = (120572 + 120591 minus 120574) exp [119899minus1

int119911

0

ℎ (120585) 120585minus1

119889120585 ] (34)

and 119902 is the best ([120572 + 120591 minus 120574] 119899)-dominant

Proof Let 119866(119911) isin 119876(120572 120591 120574 120573) then

119866 (119911) = (120572 + 120591 minus 120574) +

infin

sum120581=2

120582120581[ [120573+120581minus1]119875

120485+1] 119886120581119911120581minus1

120573119875120485+1

(35)

Since 119866(119911) is analytic in 119880 and 119866(0) = 120572 + 120591 minus 120574 it suffices toshow that

1199111198661015840

(119911)

119866 (119911) ≺ ℎ (119911) (36)

Following the same argument in [10] (pages 76 and 77) (36)is true Application of Lemma 5 provesTheorem 13 with 119902(119911)

as the best ([120572 + 120591 minus 120574] 119899)-dominant

Example 14 Let ℎ(119911) = radic119911(1 minus 119911) if

1199111198661015840 (119911)

119866 (119911) ≺

radic119911

1 minus 119911(37)

then

119866 (119911) ≺ (120572 + 120591 minus 120574) 1 + radic119911

1 minus radic119911

1119899

120572 + 120591 = 120574 (38)

and 119902 is the best (120572 + 120591 minus 120574 119899)-dominant

Solution If ℎ(119911) = radic119911(1 minus 119911) and 119911 = 119903119890119894120579 0 lt 120579 lt 120587 thensimple calculation shows that ℎ(0) = 0 and ℎ(119911) is starlike andthe argument in [10] shows (37) The proof also follows fromLemma 5

Theorem 15 Let 119866(119911) isin 119876(120485)(120572 120591 0 120573) and minus1 le 119861 lt 119860 le 1

120573 gt 120485 + 1 with 120572 + 120591 = 0 If

119866 (119911) ≺ (120572 + 120591)1 + 119860119911

1 + 119861119911 (39)

then[119865(119911)]

(120485)

[119911120573](120485)

≺(120572 + 120591) (120573 minus 120485)

120591120581119911(120572+120591)(120485minus120573)120591120581

times int119911

0

120585(120572(120573minus120485)+120591(120573minus120485minus120581))120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(40)

Proof Let 119866(119911) isin 119876(120485)(120572 120591 0 120573) and from (6)

1

(120572 + 120591)119866 (119911) = 1 +

infin

sum120581=2

119862120581119886120581119911120581minus1

(41)

where

119862120581=

120582120581[ (120573+120581minus1)119875

120485+1] sdot [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591) [120573

119875120485+1

] (42)

Let 119892(119911) = [119865(119911)](120485)

[119911120573](120485) then 119892(119911) = 1 + sum

infin

120581=2120603120581119886120581119911120581minus1

where 120603120581= 120582120581[(120573+120581minus1)

119875120485+1

][(120573)

119875120485] and from (39) we have

119892 (119911) +120591

(120572 + 120591) (120573 minus 120485)1199111198921015840

(119911) ≺ (1 + 119860119911

1 + 119861119911) = ℎ (119911) (43)

and ℎ(119911) is convex and univalent in 119880 So by Lemma 6

119892 (119911) ≺(120572 + 120591) (120573 minus 120485)

120591120581119911minus(120572+120591)(120573minus120485)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(44)

This completes the proof of Theorem 15

Corollary 16 Let

119876(0)

(120572 120591 0 120573) =120572 (119865 (119911))

119911120573+

120591[119865 (119911)]1015840

[119911120573]1015840

(45)

If

119876(0)

(120572 120591 0 120573) ≺ (120572 + 120591) (1 + 119860119911

1 + 119861119911) (46)

then

119865 (119911)

119911120573≺

120572120573 + 120591 (120573 minus 120581)

120591120581119911minus120573(120572+120591)120591120581

times int119911

0

120585(120573(120572+120591)minus120591120581)120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(47)

Proof The result follows fromTheorem 15



Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

The work here is fully supported by LRGSTD2011UKMICT0302 and GUP-2013-004 The authors also would liketo thank the referee and editor in charge for the commentsand suggestions given to improve their paper


References

[1] H Saitoh ldquoOn inequalities for certain analytic functionsrdquoMath-ematica Japonica vol 35 no 6 pp 1073ndash1076 1990

[2] S Owa ldquoSome properties of certain analytic functionsrdquo Soo-chow Journal of Mathematics vol 13 no 2 pp 197ndash201 1987

[3] S Owa ldquoGeneralization properties for certain analytic func-tionsrdquo International Journal of Mathematics and MathematicalSciences vol 21 no 4 pp 707ndash712 1998

[4] Z Wang C Gao and S Yuan ldquoOn the univalency of certainanalytic functionsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 1 pp 1ndash4 2006

[5] HHayami S Owa andHM Srivastava ldquoCoefficient estimatesfor a certain class of analytic functions involving the argumentsof their derivativerdquo Jnanabha vol 43 pp 37ndash43 2013

[6] Q-H Xu H-G Xiao and HM Srivastava ldquoSome applicationsof differential sub-ordination and the Dziok-Srivastava convo-lution operatorrdquo Applied Mathematics and Computation vol230 pp 496ndash508 2014

[7] L F Stanciu D Breaz and H M Srivastava ldquoSome criteria forunivalence of a certain integral operatorrdquo Novi Sad Journal ofMathematics vol 43 pp 51ndash57 2013

[8] D J Hallenbeck and T H MacGregor Linear Problems andConvexity Techniques in Geometric Function Theory vol 22Pitman Boston Mass USA 1984

[9] D J Hallenbeck ldquoConvex hulls and extreme points of somefamilies of univalent functionsrdquo Transactions of the AmericanMathematical Society vol 192 pp 285ndash292 1974

[10] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[11] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981

[12] AWGoodmanUnivalent Functions vol 1 Polygon PublishingHouse Washington DC USA 1983

Research ArticleA Study of Cho-Kwon-Srivastava Operator withApplications to Generalized Hypergeometric Functions

F Ghanim1 and M Darus2

1 Department of Mathematics College of Sciences University of Sharjah Sharjah UAE2 School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia43600 Bangi Selangor Malaysia

Correspondence should be addressed to M Darus maslinaukmedumy

Received 16 May 2014 Accepted 21 June 2014 Published 9 July 2014


Copyright copy 2014 F Ghanim and M Darus This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce a new class ofmeromorphically analytic functions which is defined bymeans of aHadamard product (or convolution)involving some suitably normalized meromorphically functions related to Cho-Kwon-Srivastava operator A characterizationproperty giving the coefficient bounds is obtained for this class of functionsThe other related properties which are investigated inthis paper include distortion and the radii of starlikeness and convexity We also consider several applications of our main resultsto generalized hypergeometric functions

1 Introduction

A meromorphic function is a single-valued function that isanalytic in all but possibly a discrete subset of its domain andat those singularities it must go to infinity like a polynomial(ie these exceptional points must be poles and not essentialsingularities) A simpler definition states that a meromorphicfunction 119891(119911) is a function of the form

119891 (119911) =119892 (119911)

ℎ (119911) (1)

where 119892(119911) and ℎ(119911) are entire functions with ℎ(119911) = 0 (see[1 page 64]) A meromorphic function therefore may onlyhave finite-order isolated poles and zeros and no essentialsingularities in its domain An equivalent definition of ameromorphic function is a complex analytic map to theRiemann sphere For example the gamma function is mero-morphic in the whole complex plane C

In the present paper we initiate the study of functionswhich are meromorphic in the punctured disk 119880lowast = 119911 0 lt

|119911| lt 1 with a Laurent expansion about the origin see [2]

Let119860 be the class of analytic functions ℎ(119911)with ℎ(0) = 1which are convex and univalent in the open unit disk 119880 =

119880lowast cup 0 and for which

R ℎ (119911) gt 0 (119911 isin 119880lowast

) (2)

For functions 119891 and 119892 analytic in 119880 we say that 119891 is subor-dinate to 119892 and write

119891 ≺ 119892 in 119880 or 119891 (119911) ≺ 119892 (119911) (119911 isin 119880lowast

) (3)

if there exists an analytic function 119908(119911) in 119880 such that

|119908 (119911)| le |119911| 119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880lowast

) (4)

Furthermore if the function 119892 is univalent in 119880 then

119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0)

119891 (119880) sube 119892 (119880) (119911 isin 119880lowast

) (5)

This paper is divided into two sections the first intro-duces a new class of meromorphically analytic functionswhich is defined by means of a Hadamard product (or



convolution) involving linear operator The second sectionhighlights some applications of the main results involvinggeneralized hypergeometric functions

2 Preliminaries

Let Σ denote the class of meromorphic functions 119891(119911) nor-malized by

119891 (119911) =1

119911+

infin

sum119899=1

119886119899119911119899

(6)

which are analytic in the punctured unit disk 119880lowast = 119911 0 lt

|119911| lt 1 For 0 le 120573 we denote by 119878lowast(120573) and 119896(120573) the sub-classes of Σ consisting of all meromorphic functions whichare respectively starlike of order 120573 and convex of order 120573 in119880

For functions 119891119895(119911) (119895 = 1 2) defined by

119891119895(119911) =

1

119911+

infin

sum119899=1

119886119899119895119911119899

(7)

we denote the Hadamard product (or convolution) of 1198911(119911)

and 1198912(119911) by

(1198911lowast 1198912) =

1

119911+

infin

sum119899=1

11988611989911198861198992119911119899

(8)

Cho et al [3] andGhanimandDarus [4] studied the followingfunction

119902120582120583

(119911) =1

119911+

infin

sum119899=1

(120582

119899 + 1 + 120582)

120583

119911119899

(120582 gt 0 120583 ge 0) (9)

Corresponding to the function 119902120582120583(119911) and using the

Hadamard product for 119891(119911) isin Σ we define a new linearoperator 119871(120582 120583) on Σ by

119871120582120583119891 (119911) = (119891 (119911) lowast 119902

120582120583(119911))

=1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(10)

TheHadamard product or convolution of the functions119891given by (10) with the functions 119871

119905119886119892 and 119871

119905119886ℎ given respec-

tively by

119871120582120583119892 (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast 119892 (119911) isin Σ)

119871120582120583ℎ (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198881198991003816100381610038161003816 119911119899

(119911 isin 119880lowast

ℎ (119911) isin Σ)

(11)

can be expressed as follows

119871120582120583

(119891 lowast 119892) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast)

119871120582120583

(119891 lowast ℎ) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198881198991003816100381610038161003816 119911119899


)

(12)

By applying the subordination definition we introducehere a new classΣ120583

120582(120588 119860 119861) ofmeromorphic functions which

is defined as follows

Definition 1 A function 119891 isin Σ of the form (6) is said to be inthe class Σ120583

120582(120588 119860 119861) if it satisfies the following subordination

property

120588119871120582120583

(119891 lowast 119892) (119911)

119871120582120583

(119891 lowast ℎ) (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911 (119911 isin 119880

lowast

) (13)

where minus1 le 119861 lt 119860 le 1 120588 gt 0 with condition 0 le |119888119899| le |119887119899|

and 119871(120582 120583)(119891 lowast ℎ)(119911) = 0

As for the second result of this paper on applicationsinvolving generalized hypergeometric functions we need toutilize the well-known Gaussian hypergeometric functionOne denotes 120601(120572 120573 119911) the class of the function given by

120601 (120572 120573 119911) =1

119911+

infin

sum119899=0

(120572)119899+1

(120573)119899+1

119911119899

(14)

for120573 = 0 minus1 minus2 and120572 isin C0 where (120582)119899 = 120582(120582 + 1)119899+1

is the Pochhammer symbol We note that

120601 (120572 120573 119911) =1

119911 21198651(1 120572 120573 119911) (15)

where

21198651(119887 120572 120573 119911) =

infin

sum119899=0

(119887)119899(120572)119899

(120573)119899

119911119899

119899(16)

is the well-known Gaussian hypergeometric functionCorresponding to the functions 120601(120572 120573 119911) and 119902

120582120583(119911)

given in (9) and using the Hadamard product for 119891(119911) isin Σwe define a new linear operator 119871(120572 120573 120582 120583) on Σ by

119871 (120572 120573 120582 120583) 119891 (119911) = (119891 (119911) lowast 120601 (120572 120573 119911) lowast 119902120582120583

(119911))

=1

119911+

infin

sum119899=1

(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(17)

The meromorphic functions with the generalized hypergeo-metric functions were considered recently by Cho and Kim[5] Dziok and Srivastava [6 7] Ghanim [8] Ghanim et al[9 10] and Liu and Srivastava [11 12]


Now it follows from (17) that

119911(119871 (120572 120573 120582 120583) 119891 (119911))1015840

= 120572119871 (120572 + 1 120573 120582 120583) 119891 (119911)

minus (120572 + 1) 119871 (120572 120573 120582 120583) 119891 (119911) (18)

The subordination relation (13) in conjunction with (17)takes the following form

120588119871 (120572 + 1 120573 120582 120583) 119891 (119911)

119871 (120572 120573 120582 120583) 119891 (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911

(0 le 119861 lt 119860 le 1 120588 gt 0)

(19)

Definition 2 A function 119891 isin Σ of the form (6) is said to bein the class Σ120583

120582(120588 120572 120573 119860 119861) if it satisfies the subordination

relation (19) above

3 Characterization and OtherRelated Properties

In this section we begin by proving a characterizationproperty which provides a necessary and sufficient conditionfor a function 119891 isin Σ of the form (6) to belong to the classΣ120583

120582(120588 119860 119861) of meromorphically analytic functions

Theorem 3 The function 119891 isin Σ is said to be a member of theclass Σ120583

120582(120588 119860 119861) if and only if it satisfies

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861)

minus10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(20)

The equality is attained for the function 119891119899(119911) given by

119891119899(119911) =

1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

(21)

Proof Let 119891 of the form (6) belong to the class Σ120583120582(120588 119860 119861)

Then in view of (12) we find that

1003816100381610038161003816100381610038161003816100381610038161003816

120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899) 119911119899+1

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

times (120588119861119887119899+ 119888119899(119860 minus 119861) minus 120588119861) 119886

119899119911119899+1

)

minus11003816100381610038161003816100381610038161003816100381610038161003816

le 120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899)10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899

10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816)

minus1

le 1

(22)

Putting |119911| = 119903 (0 le 119903 lt 1) and noting the fact thatthe denominator in the above inequality remains positive byvirtue of the constraints stated in (13) for all 119903 isin [0 1) weeasily arrive at the desired inequality (20) by letting 119911 rarr 1

Conversely if we assume that the inequality (20) holdstrue in the simplified form (22) it can readily be shown that100381610038161003816100381610038161003816100381610038161003816

120588 ((119891 lowast 119892) (119911)) minus ((119891 lowast ℎ) (119911))

120588119861 ((119891 lowast 119892) (119911)) + 120588 (119860 minus 119861) minus 120588119861 ((119891 lowast ℎ) (119911))

100381610038161003816100381610038161003816100381610038161003816lt 1

(119911 isin 119880lowast)

(23)

which is equivalent to our condition of theorem so that 119891 isin

Σ120583

120582(120588 119860 119861) hence the theorem

Theorem 3 immediately yields the following result

Corollary 4 If the function 119891 isin Σ belongs to the classΣ120583

120582(120588 119860 119861) then

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119899 ge 1

(24)

where the equality holds true for the functions 119891119899(119911) given by

(21)

We now state the following growth and distortion prop-erties for the class Σ120583

120582(120588 119860 119861)

Theorem 5 If the function 119891 defined by (6) is in the classΣ120583

120582(120588 119860 119861) then for 0 lt |119911| = 119903 lt 1 one has

1

119903minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903

le1003816100381610038161003816119891 (119911)

1003816100381610038161003816

le1

119903+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903


1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

le100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(25)

Proof Since 119891 isin Σ120583

120582(120588 119860 119861) Theorem 3 readily yields the

inequalityinfin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(26)

Thus for 0 lt |119911| = 119903 lt 1 and utilizing (26) we have

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

le1

119903+ 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

119903+ 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|minus

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

ge1

119903minus 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

ge1

119903minus 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(27)

Also fromTheorem 3 we getinfin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(119860 minus 119861) (2 + 120582)

120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(28)

Hence

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2+

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

le1

1199032+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

ge1

1199032minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816

ge1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(29)


We next determine the radii of meromorphic starlikenessand meromorphic convexity of the class Σ120583

120582(120588 119860 119861) which

are given byTheorems 6 and 7 below


120582(120588 119860 119861) then 119891 is meromorphic starlike of order 120575 in the

disk |119911| lt 1199031 where

1199031= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times ((119860 minus 119861) (119899 + 2 minus 120575))minus1

1(119899+1)

(30)

The equality is attained for the function 119891119899(119911) given by (21)

Proof It suffices to prove that1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (31)

For |119911| lt 1199031 we have

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 1 + 120582))

120583

119886119899119911119899

1119911 + suminfin

119899=1(120582(119899 + 1 + 120582))

120583

119886119899119911119899

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

100381610038161003816100381610038161003816100381610038161003816

lesuminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(32)

Hence (32) holds true forinfin

sum119899=1

(119899 + 1) (120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

le (1 minus 120575) (1 minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

)

(33)

or

suminfin

119899=1(119899 + 2 minus 120575) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(1 minus 120575)le 1 (34)

With the aid of (20) and (34) it is true to say that for fixed 119899

(119899 + 2 minus 120575) (120582(119899 + 1 + 120582))120583

|119911|119899+1

(1 minus 120575)

le (120582

119899 + 120582 + 1)

120583

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times (119860 minus 119861)minus1

119899 ge 1

(35)


Solving (35) for |119911| we obtain

|119911| le (1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times ((119899 + 2 minus 120575) (119860 minus 119861))minus1

119899+1

(36)



120582(120588 119860 119861) then 119891 is meromorphic convex of order 120575 in the

disk |119911| lt 1199032 where

1199032= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times (119899 (119899 + 2 minus 120575) (119860 minus 119861))minus1

1(119899+1)

(37)


Proof By using the same technique employed in the proof ofTheorem 6 we can show that

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891 (119911))10158401015840

(119891 (119911))1015840+ 2

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (38)

For |119911| lt 1199031and with the aid of Theorem 3 we have the

assertion of Theorem 7

4 Applications Involving GeneralizedHypergeometric Functions


120582(120588 120572 120573 119860 119861) if and only if it satisfies

infin

sum119899=1

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(39)


119891119899(119911)

=1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

119899 ge 1

(40)

Proof By using the same technique employed in the proof ofTheorem 3 along with Definition 2 we can proveTheorem 8

The following consequences of Theorem 8 can bededuced by applying (39) and (40) along with Definition 2


120582(120588 120572 120573 119860 119861) then1003816100381610038161003816119886119899

1003816100381610038161003816

le(119860 minus 119861) (119899 + 120582 + 1)

120583

(120573)119899+1

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861)) (120572)

119899+1

119899 ge 1

(41)


(40)

Corollary 10 If the function 119891 defined by (6) is in the classΣ120583

120582(120588 120572 120573 119860 119861) then 119891 is meromorphic starlike of order 120575 in

the disk |119911| lt 1199033 where

1199033= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(42)



120582(120588 120572 120573 119860 119861) then 119891 is meromorphic convex of order 120575 in


1199034= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(43)





All authors read and approved the final paper

Acknowledgment

The work here was fully supported by FRGSTOPDOWN2013ST06UKM011

References

[1] S G Krantz ldquoMeromorphic functions and singularities at infin-ityrdquo inHandbook of Complex Variables pp 63ndash68 BirkhaauserBoston Mass USA 1999


[2] A W Goodman ldquoFunctions typically-real and meromorphicin the unit circlerdquo Transactions of the American MathematicalSociety vol 81 pp 92ndash105 1956

[3] N E Cho O S Kwon and H M Srivastava ldquoInclusion andargument properties for certain subclasses of meromorphicfunctions associated with a family of multiplier transforma-tionsrdquo Journal of Mathematical Analysis and Applications vol300 no 2 pp 505ndash520 2004

[4] F Ghanim and M Darus ldquoSome properties on a certain classof meromorphic functions related to Cho-Kwon-Srivastavaoperatorrdquo Asian-European Journal of Mathematics vol 5 no 4Article ID 1250052 pp 1ndash9 2012

[5] N E Cho and I H Kim ldquoInclusion properties of certainclasses of meromorphic functions associated with the gen-eralized hypergeometric functionrdquo Applied Mathematics andComputation vol 187 no 1 pp 115ndash121 2007

[6] J Dziok and H M Srivastava ldquoSome subclasses of analyticfunctions with fixed argument of coefficients associated withthe generalized hypergeometric functionrdquo Advanced Studies inContemporary Mathematics (Kyungshang) vol 5 no 2 pp 115ndash125 2002

[7] J Dziok and H M Srivastava ldquoCertain subclasses of analyticfunctions associated with the generalized hypergeometric func-tionrdquo Integral Transforms and Special Functions vol 14 no 1 pp7ndash18 2003

[8] F Ghanim ldquo A study of a certain subclass of Hurwitz-Lerch-Zeta function related to a linear operatorrdquo Abstract and AppliedAnalysis vol 2013 Article ID 763756 7 pages 2013

[9] F Ghanim and M Darus ldquoA new class of meromorphicallyanalytic functions with applications to the generalized hyper-geometric functionsrdquo Abstract and Applied Analysis vol 2011Article ID 159405 10 pages 2011

[10] F Ghanim M Darus and Z-G Wang ldquoSome properties ofcertain subclasses of meromorphically functions related to cho-kwon-srivastava operatorrdquo Information Journal vol 16 no 9pp 6855ndash6866 2013

[11] J Liu and H M Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001

[12] J Liu and H M Srivastava ldquoClasses of meromorphically multi-valent functions associated with the generalized hypergeomet-ric functionrdquoMathematical andComputerModelling vol 39 no1 pp 21ndash34 2004

Research ArticleNew Expansion Formulas for a Family of the 120582-GeneralizedHurwitz-Lerch Zeta Functions

H M Srivastava1 and Seacutebastien Gaboury2

1 Department of Mathematics and Statistics University of Victoria Victoria BC Canada V8W 3R42Department of Mathematics and Computer Science University of Quebec at Chicoutimi Chicoutimi QC Canada G7H 2B1

Correspondence should be addressed to Sebastien Gaboury s1gabouruqacca

Received 17 April 2014 Accepted 26 May 2014 Published 26 June 2014


Copyright copy 2014 H M Srivastava and S Gaboury This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We derive several new expansion formulas for a new family of the 120582-generalized Hurwitz-Lerch zeta functions which wereintroduced by Srivastava (2014) These expansion formulas are obtained by making use of some important fractional calculustheorems such as the generalized Leibniz rules the Taylor-like expansions in terms of different functions and the generalizedchain rule Several (known or new) special cases are also considered

1 Introduction

The Hurwitz-Lerch Zeta function Φ(119911 119904 119886) which is one ofthe fundamentally important higher transcendental func-tions is defined by (see eg [1 p 121 et seq] see also [2] and[3 p 194 et seq])

Φ (119911 119904 119886) =

infin

sum119899=0

119911119899

(119899 + 119886)119904

(119886 isin C Zminus

0 119904 isin C when |119911| lt 1

R (119904) gt 1 when |119911| = 1)

(1)

TheHurwitz-Lerch zeta function contains as its special casesthe Riemann zeta function 120577(119904) the Hurwitz zeta function120577(119904 119886) and the Lerch zeta function ℓ

119904(120585) defined by

120577 (119904) =

infin

sum119899=1

1

119899119904= Φ (1 119904 1) = 120577 (119904 1) (R (119904) gt 1)

120577 (119904 119886) =

infin

sum119899=0

1

(119899 + 119886)119904

= Φ (1 119904 119886) (R (119904) gt 1 119886 isin C Zminus

0)

ℓ119904(120585) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 1)119904= Φ (e2120587119894120585 119904 1) (R (119904) gt 1 120585 isin R)

(2)

respectively but also such other important functions ofAnalytic Number Theory as the Polylogarithmic function (orde Jonquierersquos function) Li

119904(119911)

Li119904(119911) =

infin

sum119899=1

119911119899

119899119904= 119911Φ (119911 119904 1)

(119904 isin C when |119911| lt 1 R (119904) gt 1 when |119911| = 1)

(3)

and the Lipschitz-Lerch zeta function 120601(120585 119886 119904) (see [1 p 122Equation 25 (11)])

120601 (120585 119886 119904) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 119886)119904= Φ (e2120587119894120585 119904 119886)


0 R (119904) gt 0 when 120585 isin R Z R (119904) gt 1

when 120585 isin Z)

(4)



Indeed the Hurwitz-Lerch zeta functionΦ(119911 119904 119886) defined in(5) can be continued meromorphically to the whole complex119904-plane except for a simple pole at 119904 = 1 with its residue 1 Itis also well known that

Φ (119911 119904 119886) =1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

1 minus 119911eminus119905d119905

(R (119886) gt 0R (119904) gt 0 when |119911| ≦ 1 (119911 = 1) R (119904) gt 1

when 119911 = 1) (5)

Motivated by the works of Goyal and Laddha [4] Lin andSrivastava [5] Garg et al [6] and other authors Srivastavaet al [7] (see also [8]) investigated various properties of anatural multiparameter extension and generalization of theHurwitz-Lerch zeta functionΦ(119911 119904 119886)defined by (5) (see also[9]) In particular they considered the following function

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) =

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

119911119899

(119899 + 119886)119904

(119901 119902 isin N0 120582

119895isin C (119895 = 1 119901)

119886 120583119895isin C Z

minus

0(119895 = 1 119902)

120588119895 120590

119896isin R

+

(119895 = 1 119901 119896 = 1 119902)

Δ gt minus1 when 119904 119911 isin C

Δ = minus1 119904 isin C when |119911| lt nablalowast

Δ = minus1 R (Ξ) gt1

2when |119911| = nabla

lowast

)

(6)

with

nablalowast

= (

119901

prod119895=1

120588minus120588119895

119895) sdot (

119902

prod119895=1

120590120590119895

119895)

Δ =

119902

sum119895=1

120590119895minus

119901

sum119895=1

120588119895

Ξ = 119904 +

119902

sum119895=1

120583119895minus

119901

sum119895=1

120582119895+119901 minus 119902

2

(7)

Here and for the remainder of this paper (120582)120581denotes

the Pochhammer symbol defined in terms of the Gammafunction by

(120582)120581=Γ (120582 + 120581)

Γ (120582)

=

120582 (120582 + 1) sdot sdot sdot (120582 + 119899 minus 1) (120581 = 119899 isin N 120582 isin C)

1 (120581 = 0 120582 isin C 0)

(8)

It is being understood conventionally that (0)0= 1 and

assumed tacitly that the Γ-quotient exists (see for details[10 p 21 et seq]) In terms of the extended Hurwitz-Lerchzeta function defined by (6) the following generalization ofseveral known integral representations arising from (5) wasgiven by Srivastava et al [7] as follows

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0)

(9)

provided that the integral exists

Definition 1 The function119897Ψlowast

119898or

119897Ψ119898(119897 119898 isin N

0) involved

in the right-hand side of (9) is the well-known Fox-Wrightfunction which is a generalization of the familiar generalizedhypergeometric function

119897119865119898(119897 119898 isin N

0) with 119897 numer-

ator parameters 1198861 119886

119897and 119898 denominator parameters

1198871 119887

119898such that

119886119895isin C (119895 = 1 119897) 119887

119895isin C Z

minus

0(119895 = 1 119898)

(10)

defined by (see for details [11 p 21 et seq] and [10 p 50 etseq])

119897Ψlowast

119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

=

infin

sum119899=0

(1198861)1198601119899sdot sdot sdot (119886

119897)119860119897119899

(1198871)1198611119899sdot sdot sdot (119887

119898)119861119898119899

119911119899

119899

=Γ (119887

1) sdot sdot sdot Γ (119887

119898)

Γ (1198861) sdot sdot sdot Γ (119886

119897) 119897Ψ119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

(119860119895gt 0 (119895 = 1 119897) 119861

119895gt 0 (119895 = 1 119898)

1 +

119898

sum119895=1

119861119895minus

119897

sum119895=1

119860119895≧ 0)

(11)

where the equality in the convergence condition holds truefor suitably bounded values of |119911| given by

|119911| lt nabla = (

119897

prod119895=1

119860minus119860119895

119895) sdot (

119898

prod119895=1

119861119861119895

119895) (12)

Recently Srivastava [12] introduced and investigated asignificantly more general class of Hurwitz-Lerch zeta type


functions by suitably modifying the integral representationformula (9) Srivastava considered the following function

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

Γ (119904)intinfin

0

119905119904minus1 exp(minus119886119905 minus 119887

119905120582)

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0)

(13)

so that clearly we have the following relationship

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 0 120582)

= Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

= e119887Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 0)

(14)

In its special case when

119901 minus 1 = 119902 = 0 (1205821= 120583 120588

1= 1) (15)

the definition (13) would reduce to the following form

Θ120582

120583(119911 119904 119886 119887)

=1

Γ (119904)intinfin

0


119905120582) (1 minus 119911eminus119905)

minus120583

d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0 120583 isin C)

(16)

where we have assumed further that

R (119904) gt 0 when 119887 = 0 |119911| ≦ 1 (119911 = 1) (17)

or

R (119904 minus 120583) gt 0 when 119887 = 0 119911 = 1 (18)

provided that the integral (16) exists The functionΘ120582

120583(119911 119904 119886 119887) was studied by Raina and Chhajed [13 Eq

(16)] and more recently by Srivastava et al [14]As a particular interesting case of the function

Θ120582

120583(119911 119904 119886 119887) we recall the following function

Θ120582

120583(119911 119904 119886 0) = Φ

lowast

120583(119911 119904 119886) =

1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

(1 minus 119911eminus119905)120583d119905

(R (119886) gt 0 R (119904) gt 0 when |119911| ≦ 1 (119911 = 1)

R (119904 minus 120583) gt 0 when 119911 = 1)

(19)

The functionΦlowast

120583(119911 119904 119886)was introduced byGoyal and Laddha

[4] as follows

Φlowast

120583(119911 119904 119886) =

infin

sum119899=0

(120583)119899

(119886 + 119899)119904

119911119899

119899 (20)

Another special case of the function Θ120582

120583(119911 119904 119886 119887) that is

worthy to mention occurs when 120582 = 120583 = 1 and 119911 = 1 Wehave

Θ1

1(1 119904 119886 119887)

= 120577119887(119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1

1 minus eminus119905exp(minus119886119905 minus 119887

119905) d119905

(21)

where the function 120577119887(119904 119886) is the extended Hurwitz zeta

function introduced by Chaudhry and Zubair [15]In his work Srivastava [12 p 1489 Eq (21)] also

derived the following series representation of the functionΦ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899

119899(120582 gt 0)

(22)

provided that both sides of (22) exist

Definition 2 The119867-function involved in the right-hand sideof (22) is the well-known Foxrsquos 119867-function defined by [16Definition 11] (see also [10 17])

119867119898119899

pq (119911)

= 119867119898119899

pq [119911 |(119886

1 119860

1) (119886p 119860p)

(1198871 119861

1) (119887q 119861q)

]

=1

2120587119894intL

Ξ (119904) 119911minus119904d119904 (119911 isin C 0

1003816100381610038161003816arg (119911)1003816100381610038161003816 lt 120587)

(23)

where

Ξ (119904) =prod

119898

119895=1Γ (119887

119895+ 119861

119895119904) sdot prod

119899

119895=1Γ (1 minus 119886

119895minus 119860

119895119904)

prodp

119895=119899+1Γ (119886

119895+ 119860

119895119904) sdot prod

q

119895=119898+1Γ (1 minus 119887

119895minus 119861

119895119904)

(24)

An empty product is interpreted as 1 119898 119899 p and q areintegers such that 1 ≦ 119898 ≦ q 0 ≦ 119899 ≦ p 119860

119895gt 0 (119895 =

1 p) 119861119895gt 0 (119895 = 1 q) 119886

119895isin C (119895 = 1 p)

119887119895isin C (119895 = 1 q) andL is a suitable Mellin-Barnes type

contour separating the poles of the gamma functions

Γ (119887119895+ 119861

119895119904)

119898

119895=1

(25)

from the poles of the gamma functions

Γ (1 minus 119886119895+ 119860

119895119904)

119899

119895=1

(26)


It is important to recall that Srivastava [12 p 1490Eq (210)] presented another series representation for thefunction Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) involving the Laguerrepolynomials 119871(120572)

119899(119909) of order 120572 and degree 119899 in 119909 generated

by (see for details [10])

(1 minus 119905)minus120572minus1 exp(minus 119909119905

1 minus 119905)

=

infin

sum119899=0

119871(120572)

119899(119909) 119905

119899

(|119905| lt 1 120572 isin C)

(27)

Explicitly it was proven by Srivastava [12 p 1490 Eq (210)]that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=eminus119887

Γ (119904)

infin

sum119899=0

119899

sum119896=0

(minus1)119896

(119899

119896) sdot Γ (119904 + 120582 (120572 + 119896 + 1))

sdot 119871(120572)

119899(119887)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 + 120582 (120572 + 119895 + 1) 119886)

(R (119886) gt 0 R (119904 + 120582120572) gt minus120582)

(28)

provided that each member of (28) exists and

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) (29)

being given by (9)Motivated by a number of recent works by the present

authors [18ndash20] and also of those of several other authors [4ndash9 21 22] this paper aims to provide many new relationshipsinvolving the new family of the 120582-generalized Hurwitz-Lerchzeta functionΦ(120588

11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

2 Pochhammer Contour IntegralRepresentation for Fractional Derivative

Themost familiar representation for the fractional derivativeof order 120572 of 119911119901119891(119911) is the Riemann-Liouville integral [23](see also [24ndash26]) that is

D120572

119911119911

119901

119891 (119911) =1

Γ (minus120572)int119911

0

119891 (120585) 120585119901

(120585 minus 119911)minus120572minus1d120585

(R (120572) lt 0 R (119901) gt 1)

(30)

where the integration is carried out along a straight line from0 to 119911 in the complex 120585-plane By integrating by part119898 timeswe obtain

D120572

119911119911

119901

119891 (119911) =d119898

d119911119898D

120572minus119898

119911119911

119901

119891 (119911) (31)

This allows us to modify the restrictionR(120572) lt 0 toR(120572) lt119898 (see [26])

Another representation for the fractional derivative isbased on the Cauchy integral formula This representation

Im(120585)

Branch line for

Branch line for

exp[minus(a + 1)ln(g(120585) minus g(z))]

exp[p(ln(g(120585)))]

C1

C2

C3

C4

Re(120585)

z

a

gminus1(0)

Figure 1 Pochhammerrsquos contour

too has beenwidely used inmany interesting papers (see egthe works of Osler [27ndash30])

The relatively less restrictive representation of the frac-tional derivative according to parameters appears to be theone based on the Pochhammerrsquos contour integral introducedby Tremblay [31 32]

Definition 3 Let 119891(119911) be analytic in a simply-connectedregion R of the complex 119911-plane Let 119892(119911) be regular andunivalent onR and let119892minus1(0) be an interior point ofRThenif 120572 is not a negative integer 119901 is not an integer and 119911 is inR 119892minus1(0) we define the fractional derivative of order 120572 of119892(119911)

119901

119891(119911) with respect to 119892(119911) by

119863120572

119892(119911)[119892 (119911)]

119901

119891 (119911)

=eminus119894120587119901Γ (1 + 120572)4120587 sin (120587119901)

times int119862(119911+119892

minus1(0)+119911minus119892

minus1(0)minus119865(119886)119865(119886))

sdot119891 (120585) [119892 (120585)]

119901

1198921015840 (120585)

[119892 (120585) minus 119892 (119911)]120572+1

d120585

(32)

For nonintegers 120572 and 119901 the functions 119892(120585)119901 and [119892(120585) minus119892(119911)]

minus120572minus1 in the integrand have two branch lines which beginrespectively at 120585 = 119911 and 120585 = 119892minus1(0) and both branches passthrough the point 120585 = 119886 without crossing the Pochhammercontour119875(119886) = 119862

1cup119862

2cup119862

3cup119862

4 at any other point as shown

in Figure 1 Here 119865(119886) denotes the principal value of theintegrand in (32) at the beginning and the ending point of thePochhammer contour 119875(119886) which is closed on the Riemannsurface of the multiple-valued function 119865(120585)

Remark 4 InDefinition 3 the function119891(119911)must be analyticat 120585 = 119892

minus1(0) However it is interesting to note here that ifwe could also allow 119891(119911) to have an essential singularity at120585 = 119892minus1(0) then (32) would still be valid

Remark 5 In case the Pochhammer contour never crossesthe singularities at 120585 = 119892minus1(0) and 120585 = 119911 in (32) then we


know that the integral is analytic for all 119901 and for all 120572 andfor 119911 in R 119892minus1(0) Indeed in this case the only possiblesingularities of119863120572

119892(119911)[119892(119911)]

119901

119891(119911) are 120572 = minus1 minus2 minus3 and119901 = 0 plusmn1 plusmn2 which can directly be identified from thecoefficient of the integral (32) However by integrating byparts 119873 times the integral in (32) by two different ways wecan show that120572 = minus1 minus2 and119901 = 0 1 2 are removablesingularities (see for details [31])

It is well known that [33 p 83 Equation (24)]

119863120572

119911119911

119901

=Γ (1 + 119901)

Γ (1 + 119901 minus 120572)119911119901minus120572

(R (119901) gt minus1) (33)

Adopting the Pochhammer based representation for thefractional derivative modifies the restriction to the case when119901 is not a negative integer

Now by using (33) in conjunction with the series repre-sentation (22) for Φ(120588

11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) we obtain thefollowing important fractional derivative formula that willplay an important role in our present investigation

119863120572

119911119911

120573minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899)

1198871120582

| (119904 1) (01

120582)]119863

120572

119911119911

120573minus1119911119899

119899

=Γ (120573)

120582Γ (119904) Γ (120573 minus 120572)

infin

sum119899=0

(120573)119899

(120573 minus 120572)119899

sdot

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899+120573minus120572minus1

119899

=Γ (120573)

Γ (120573 minus 120572)119911120573minus120572minus1

Φ(1205881120588119901112059011205901199021)

12058211205821199011205731205831120583119902120573minus120572

(119911 119904 119886 119887 120582)

(120582 gt 0 120573 minus 1 not a negative integer) (34)

3 Important Results InvolvingFractional Calculus

In this section we recall six fundamental theorems relatedto fractional calculus that will play central roles in ourwork Each of these theorems is a fundamental formularelated to the generalized chain rule for fractional derivativesthe Taylor-like expansions in terms of different types offunctions and the generalized Leibniz rules for fractionalderivatives

First of all Osler [27 p 290 Theorem 2] discovered afundamental relation fromwhich he deduced the generalized

chain rule for the fractional derivativesThis result is recalledhere as Theorem 6 below

Theorem 6 Let 119891(119892minus1(119911)) and 119891(ℎminus1(119911)) be defined andanalytic in the simply-connected region R of the complex 119911-plane and let the origin be an interior or boundary point ofR Suppose also that 119892minus1(119911) and ℎminus1(119911) are regular univalentfunctions on R and that ℎminus1(0) = 119892minus1(0) Let ∮119891(119892minus1(119911))d119911vanish over simple closed contour inRcup0 through the originThen the following relation holds true

119863120572

119892(119911)119891 (119911) = 119863

120572

ℎ(119911)119891 (119911) 119892

1015840(119911)

ℎ1015840(119911)(ℎ(119908) minus ℎ(119911)

119892(119908) minus 119892(119911))

120572+1

100381610038161003816100381610038161003816100381610038161003816119908=119911

(35)

Relation (35) allows us to obtain very easily knownand new summation formulas involving special functions ofmathematical physics

By applying the relation (35) Gaboury and Tremblay [34]proved the following corollarywhichwill be useful in the nextsection

Corollary 7 Under the hypotheses of Theorem 6 let 119901 be apositive integer Then the following relation holds true

119911119901119874120572

120573119891 (119911)

= 119901(119911119901minus1

)minus120572

sdot119911

119874120572

120573119891 (119911) (119911

119901minus1

)120572

119901minus1

prod119904=1

(1 minus119911

119908eminus(2120587119894119904)119901)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(36)

where

119892(119911)119874120572

120573sdot sdot sdot =

Γ (120573)

Γ (120572)[119892 (119911)]

1minus120573

119863120572minus120573

119892(119911)[119892 (119911)]

120572minus1

sdot sdot sdot

(37)

Next in the year 1971 Osler [35] obtained the followinggeneralized Taylor-like series expansion involving fractionalderivatives

Theorem 8 Let 119891(119911) be an analytic function in a simply-connected regionR Let 120572 and 120574 be arbitrary complex numbersand let

120579 (119911) = (119911 minus 1199110) 119902 (119911) (38)

with 119902(119911) a regular and univalent function without any zero inR Let 119886 be a positive real number and let

119870

=0 1 [119888] ([119888] 119905ℎ119890 119897119886119903119892119890119904119905 119894119899119905119890119892119890119903 119899119900119905 119892119903119890119886119905119890119903 119905ℎ119886119899 119888)

(39)

Let 119887 and 1199110be two points inR such that 119887 = 119911

0and let

120596 = exp (2120587119894119886) (40)


Then the following relationship holds true

sum119896isin119870

119888minus1

120596minus120574119896

119891 (120579minus1

(120579 (119911) 120596119896

))

=

infin

sum119899=minusinfin

[120579 (119911)]119888119899+120574

Γ (119888119899 + 120574 + 1)

sdot119863119888119899+120574

119911minus119887119891 (119911) 120579

1015840

(119911) (119911minus119911

0

120579 (119911))

119888119899+120574+1

100381610038161003816100381610038161003816100381610038161003816119911=1199110

(1003816100381610038161003816119911minus1199110

1003816100381610038161003816=100381610038161003816100381611991101003816100381610038161003816)

(41)

In particular if 0 lt 119888 ≦ 1 and 120579(119911) = (119911 minus 1199110) then 119896 = 0

and the formula (41) reduces to the following form

119891 (119911) = 119888

infin


(119911 minus 1199110)119888119899+120574

Γ(119888119899 + 120574 + 1)119863

119888119899+120574

119911minus119887119891(119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199110

(42)

This last formula (42) is usually referred to as the Taylor-Riemann formula and has been studied in several papers[29 36ndash39]

We next recall that Tremblay et al [40] discovered thepower series of an analytic function 119891(119911) in terms of therational expression ((119911minus119911

1)(119911minus119911

2)) where 119911

1and 119911

2are two

arbitrary points inside the regionR of analyticity of 119891(119911) Inparticular they obtained the following result

Theorem 9 (i) Let 119888 be real and positive and let

120596 = exp(2120587119894119886) (43)

(ii) Let 119891(119911) be analytic in the simply-connected region Rwith 119911

1and 119911

2being interior points of R (iii) Let the set of

curves

119862 (119905) 119862 (119905) subR 0 lt 119905 ≦ 119903 (44)

be defined by

119862 (119905) = 1198621(119905) cup 119862

2(119905)

= 119911 1003816100381610038161003816120582119905 (1199111 1199112 119911)

1003816100381610038161003816 =1003816100381610038161003816100381610038161003816120582119905(119911

1 119911

21199111+ 119911

2

2)1003816100381610038161003816100381610038161003816

(45)

where

120582119905(119911

1 119911

2 119911) = [119911 minus

1199111+ 119911

2

2+ 119905 (

1199111minus 119911

2

2)]

sdot [119911 minus (1199111+ 119911

2

2) minus 119905 (

1199111minus 119911

2

2)]

(46)

which are the Bernoulli type lemniscates (see Figure 2) withcenter located at (119911

1+ 119911

2)2 and with double-loops in which

one loop 1198621(119905) leads around the focus point

1199111+ 119911

2

2+ (

1199111minus 119911

2

2) 119905 (47)

and the other loop 1198622(119905) encircles the focus point1199111+ 119911

2

2minus (

1199111minus 119911

2

2) 119905 (48)

Im(120585)

Branch line for

Branch line for

C1

C2

z1

z2

Re(120585)

z

c =z1 + z2

2

(120585 minus z2)120583120579(120585)

aminus120574minus1

(120585 minus z1)120579(120585)

aminus120574minus1

Figure 2 Multiloops contour

for each 119905 such that 0 lt 119905 ≦ 119903 (iv) Let

[(119911 minus 1199111) (119911 minus 119911

2)]

120582

= exp (120582 ln (120579 ((119911 minus 1199111) (119911 minus 119911

2))))

(49)

denote the principal branch of that function which is continu-ous and inside 119862(119903) cut by the respective two branch lines 119871

plusmn

defined by

119871plusmn=

119911 119911 =1199111+ 119911

2

2plusmn 119905 (

1199111minus 119911

2

2) (0 ≦ 119905 ≦ 1)

119911 119911 =1199111+ 119911

2

2plusmn 119894119905 (

1199111minus 119911

2

2) (119905 lt 0)

(50)

such that ln((119911minus1199111)(119911minus119911

2)) is real when (119911minus119911

1)(119911minus119911

2) gt 0 (v)

Let 119891(119911) satisfy the conditions of Definition 3 for the existenceof the fractional derivative of (119911 minus 119911

2)119901

119891(119911) of order 120572 for 119911 isinR 119871

+cup 119871

minus denoted by119863120572

119911minus1199112

(119911 minus 1199112)119901

119891(119911) where 120572 and119901 are real or complex numbers (vi) Let

119870 = 119896 119896 isin N arg(120582119905(119911

1 119911

21199111+ 119911

2

2))

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) +

2120587119896

119886

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) + 2120587

(51)

Then for arbitrary complex numbers 120583]120574 and for 119911 on 1198621(1)

defined by

120585 =1199111+ 119911

2

2+1199111minus 119911

2

2radic1 + e119894120579 (minus120587 lt 120579 lt 120587)

sum119896isin119870

119888minus1120596minus120574119896

1199111minus 119911

2

119891 (120601minus1

(120596119896

120601 (119911)))

times [120601minus1

(120596119896

120601(119911)) minus 1199111]][120601

minus1

(120596119896

120601(119911)) minus 1199112]120583

=

infin


e119894120587119888(119899+1)sin [(120583 + 119888119899 + 120574) 120587]sin [(120583 minus 119888 + 120574) 120587] Γ (1 minus ] + 119888119899 + 120574)

sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

[120601 (119911)]119888119899+120574

(52)


where

120601 (119911) =119911 minus 119911

1

119911 minus 1199112

(53)

The case 0 lt 119888 ≦ 1 of Theorem 9 reduces to the followingform

119888minus1119891 (119911) (119911 minus 1199111)](119911 minus 119911

2)120583

1199111minus 119911

2

=

infin


e119894120587119888(119899+1) sin [(120583 + 119888119899 + 120574) 120587]sin [(120583 minus 119888 + 120574) 120587] Γ (1 minus ] + 119888119899 + 120574)

sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

(119911 minus 119911

1

119911 minus 1199112

)

119888119899+120574

(54)

Tremblay and Fugere [41] developed the power seriesof an analytic function 119891(119911) in terms of the function (119911 minus1199111)(119911 minus 119911

2) where 119911

1and 119911

2are two arbitrary points inside

the analyticity region R of 119891(119911) Explicitly they gave thefollowing theorem

Theorem 10 Under the assumptions ofTheorem 9 the follow-ing expansion formula holds true

sum119896isin119870

119888minus1

120596minus120574119896

[(1199112minus 119911

1+ radicΔ

119896

2)

120572

(1199111minus 119911

2+ radicΔ

119896

2)

120573

sdot 119891 (1199111+ 119911

2+ radicΔ

119896

2)

minus e119894120587(120572minus120573)sin [(120572 + 119888 minus 120574) 120587]sin [(120573 + 119888 minus 120574) 120587]

sdot (1199112minus 119911

1minus radicΔ

119896

2)

120572

times(1199111minus 119911

2minus radicΔ

119896

2)

120573

119891(1199111+ 119911

2minus radicΔ

119896

2)]

=

infin


sin [(120573 minus 119888119899 minus 120574) 120587]sin [(120573 minus 119888 minus 120574) 120587]

eminus119894120587119888(119899+1)[120579 (119911)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911minus1199112

times(119911minus1199112)120573minus119888119899minus120574minus1

(120579 (119911)

(119911minus1199112) (119911minus119911

1))

minus119888119899minus120574minus1

1205791015840

(119911)119891 (119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199111

(55)

where

Δ119896= (119911

1minus 119911

2)2

+ 4119881 (120596119896

120579 (119911))

119881 (119911) =

infin

sum119903=1

119863119903minus1

119911[119902 (119911)]

minus119903

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

119911119903

119903

120579 (119911) = (119911 minus 1199111) (119911 minus 119911

2) 119902 ((119911 minus 119911

1) (119911 minus 119911

2))

(56)

As a special case if we set 0 lt 119888 ≦ 1 119902(119911) = 1 (120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2)) and 119911

2= 0 in (55) we obtain

119891 (119911) = 119888119911minus120573

(119911 minus 1199111)minus120572

times

infin


sin [(120573 minus 119888119899 minus 120574) 120587]sin [(120573 + 119888 minus 120574) 120587]

e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574minus1

(119911 + 119908 minus 1199111)119891(119911)

10038161003816100381610038161003816119911=1199111

(119908=119911)

(57)

Finally we give two generalized Leibniz rules for frac-tional derivatives Theorem 11 is a slightly modified theoremobtained in 1970 by Osler [28] Theorem 12 was givensome years ago by Tremblay et al [42] with the help ofthe properties of Pochhammerrsquos contour representation forfractional derivatives

Theorem 11 (i) Let R be a simply-connected region contain-ing the origin (ii) Let 119906(119911) and V(119911) satisfy the conditions ofDefinition 3 for the existence of the fractional derivative ThenforR(119901 + 119902) gt minus1 and 120574 isin C the following Leibniz rule holdstrue

119863120572

119911119911

119901+119902

119906 (119911) V (119911)

=

infin


(120572

120574 + 119899)119863

120572minus120574minus119899

119911119911

119901

119906 (119911)119863120574+119899

119911119911

119902V (119911) (58)

Theorem 12 (i) LetR be a simply-connected region contain-ing the origin (ii) Let 119906(119911) and V(119911) satisfy the conditions ofDefinition 3 for the existence of the fractional derivative (iii)Let U sub R be the region of analyticity of the function 119906(119911)and letV subR be the region of analyticity of the function V(119911)Then for

119911 = 0 119911 isin U capV R (1 minus 120573) gt 0 (59)

the following product rule holds true

119863120572

119911119911

120572+120573minus1

119906 (119911) V (119911)

=119911Γ (1 + 120572) sin (120573120587) sin (120583120587) sin [(120572 + 120573 minus 120583) 120587]sin [(120572 + 120573) 120587] sin [(120573 minus 120583 minus ]) 120587] sin [(120583 + ]) 120587]

sdot

infin


119863120572+]+1minus119899119911

119911120572+120573minus120583minus1minus119899119906 (119911)119863minus1minus]+119899119911

119911120583minus1+119899V (119911)Γ (2 + 120572 + ] minus 119899) Γ (minus] + 119899)

(60)

4 Main Expansion Formulas

This section is devoted to the presentation of the new relationsinvolving the new family of the 120582-generalized Hurwitz-Lerchzeta functionΦ(120588

11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Theorem 13 Under the hypotheses of Corollary 7 let 119896 be apositive integer Then the following relation holds true

Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 (1120582))]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot 119865(119896minus1)

119863[119896120572 + 119899 1 + 120572 minus 120573 1 + 120572 minus 120573 120573 + (119896 minus 1) 120572 + 119899

eminus2120587119894119896 eminus2(119896minus1)120587119894119896] (61)

where 120582 gt 0 and 119865(119899)119863

denotes the Lauricella function of 119899variables defined by [11 p 60]

119865(119899)

119863[119886 119887

1 119887

119899 119888 119909

1 119909

119899]

=

infin

sum1198981119898119899=0

(119886)1198981+sdotsdotsdot+119898

119899

(1198871)1198981

sdot sdot sdot (119887119899)119898119899

(119888)1198981+sdotsdotsdot+119898

119899

1199091198981

1

1198981sdot sdot sdot

119909119898119899

119899

119898119899

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091198991003816100381610038161003816 lt 1)

(62)


Proof Putting 119901 = 119896 and letting 119891(119911) =

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Corollary 7 we get

119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119896

(119911119896minus1)120572

sdot119911119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) (119911119896minus1

)120572

sdot

119896minus1

prod119904=1

(1 minus119911

119908eminus2120587119894119904119896)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(63)

With the help of the definition of119911119874120572

120573given by (37) we find

for the left-hand side of (63) that

119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

(64)

We now expand each factor in the product in (63) inpower series and replace the generalized Hurwitz-Lerch zeta

function by its 119867-function series representation We thusfind for the right-hand side of (63) that

119896

(119911119896minus1)120572

119911

119874120572

120573

times

1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119911119899+(119896minus1)120572

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981

sdot sdot sdot (1 + 120572 minus 120573)119898119896minus1

sdot((119911119908) eminus2120587119894119896)

1198981

1198981

sdot sdot sdot((119911119908) eminus2(119896minus1)120587119894119896)

119898119896minus1

119898119896minus1

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119908=119911

=119896

120582Γ (119904) (119911119896minus1)120572

times

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


1199111198981+sdotsdotsdot+119898119896minus1

sdot(eminus2120587119894119896)

1198981

1198981

sdot sdot sdot(eminus2(119896minus1)120587119894119896)

119898119896minus1

119898119896minus1

times119911

119874120572

120573119911

1198981+sdotsdotsdot+119898

119896minus1+119899+(119896minus1)120572

=119896

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


times(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

sdotΓ (120573) Γ (119898


119896minus1+ 119899 + 119896120572)

Γ (120572) Γ (120573 + 1198981+ sdot sdot sdot + 119898

119896minus1+ 119899 + 119896120572)


=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 1120582)]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(119899+119896120572)1198981+sdotsdotsdot+119898

119896minus1

(1+120572minus120573)1198981

sdot sdot sdot (1+120572minus120573)119898119896minus1

(120573 + (119896 minus 1) 120572)1198981+sdotsdotsdot+119898

119896minus1

sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

(65)

Finally by combining (64) and (65) we obtain the result(61) asserted byTheorem 13

We now shift our focus on the different Taylor-likeexpansions in terms of different types of functions involvingthe new family of the 120582-generalized Hurwitz-Lerch zetafunctionΦ(120588

11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


119911minus1198881198990(119911 minus 119911

0)119888119899

Γ (119888119899 + 1) Γ (1 minus 119888119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(1003816100381610038161003816119911 minus 1199110

1003816100381610038161003816 =100381610038161003816100381611991101003816100381610038161003816 120582 gt 0)

(66)

provided that both members of (66) exist

Proof Setting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 8 with 119887 = 120574 = 0 0 lt 119888 ≦ 1 and 120579(119911) = 119911 minus 119911

0 we

have

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


(119911 minus 1199110)119888119899

Γ (1 + 119888119899)

sdot119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

(67)

for 1199110= 0 and for 119911 such that |119911 minus 119911

0| = |119911

0|

Now by making use of (34) with 120573 = 1 and 120572 = 119888119899 wefind that

119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

=119911minus1198881198990

Γ (1 minus 119888119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(68)

By combining (67) and (68) we get the result (66) asserted byTheorem 14

Theorem 15 Under the hypotheses ofTheorem 9 the followingexpansion formula holds true

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin


e119894120587119888(119899+1) sin [(120572 + 119888119899 + 120574) 120587] Γ (120572 + 119888119899 + 120574)sin [(120572 minus 119888 + 120574) 120587] Γ (1 minus 120573 + 119888119899 + 120574) Γ (120572 + 120573)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(69)

for 120582 gt 0 and for 119911 on 1198621(1) defined by

119911 =1199111

2+1199111

2radic1 + e119894120579 (minus120587 lt 120579 lt 120587) (70)


Proof By taking 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 9 with 119911

2= 0 120583 = 120572 ] = 120573 and 0 lt 119888 ≦ 1 we find

that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888(119911 minus 1199111)minus120573

119911minus120572

1199111

sdot

infin


e119894120587119888(119899+1) sin [(120572 + 119888119899 + 120574) 120587]sin [(120572 minus 119888 + 120574) 120587] Γ (1 minus 120573 + 119888119899 + 120574)

sdot (119911 minus 119911

1

119911)119888119899+120574

sdot119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

(71)

Now with the help of the relation (34) with 120572997891rarr minus120573 + 119888119899 + 120574

and 120573997891rarr 120572 + 119888119899 + 120574 minus 1 we have

119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120572+120573minus1

1

Γ (120572 + 119888119899 + 120574)

Γ (120572 + 120573)

times Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582)

(72)

Thus by combining (71) and (72) we are led to the assertion(69) of Theorem 15


Theorem 16 Under the hypotheses of Theorem 10 the follow-ing expansion formula holds true

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573+120574

(119911 minus 1199111)minus120572+120574

119911120573+120572minus2120574minus1

1

sdot

infin


sin [(120573 minus 119888119899 minus 120574) 120587] e119894120587119888(119899+1)

sin [(120573 + 119888 minus 120574) 120587] Γ (1 minus 120572 + 119888119899 + 120574)

sdot (119911(119911 minus 119911

1)

11991121

)

119888119899

Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdot [(119911 minus 1199111)Φ

(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (120573 minus 119888119899 minus 120574

120572 + 120573 minus 2119888119899 minus 2120574) 119911

1

sdotΦ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)]

(73)


119911 =1199111

2+1199111

2radic1 + ei120579 (minus120587 lt 120579 lt 120587) (74)


Proof Putting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 10 with 119911

2= 0 0 lt 119888 ≦ 1 119902(119911) = 1 and 120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2) we find that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573

(119911 minus 1199111)minus120572

sdot

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911

times119911120573minus119888119899minus120574minus1

(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

(75)

With the help of the relations in (34) we have

119863minus120572+119888119899+120574

119911

times 119911120573minus119888119899minus120574minus1

(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

= 119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

+ (119911 minus 1199111)119863

minus120572+119888119899+120574

119911


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120573+120572minus2119888119899minus2120574

1

times (Γ (1 + 120573 minus 119888119899 minus 120574)

Γ (1 + 120573 + 120572 minus 2119888119899 minus 2120574)

sdot Φ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (119911 minus 119911

1

1199111

)Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdotΦ(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582) )

(76)

Thus by combining (75) and (76) we obtain the desired result(73)

Finally from the two generalized Leibniz rules forfractional derivatives given in Section 3 we obtain thefollowing two expansion formulas involving the newfamily of the 120582-generalized Hurwitz-Lerch zeta functionΦ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

=Γ (120591) Γ (1 + ] minus 120591) sin (120574120587)

120587

sdot

infin


(minus1)119899

Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(120574 + 119899) Γ (1 + ] minus 120591 minus 120574 minus 119899) Γ (120591 + 120574 + 119899)

(77)


Proof Setting 119906(119911) = 119911]minus1 and V(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886119887 120582) in Theorem 11 with 119901 = 119902 = 0 and 120572 = ] minus 120591 we obtain

119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=

infin


(] minus 120591120574 + 119899

)119863]minus120591minus120574minus119899119911

119911]minus1

sdot 119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(78)

which with the help of (33) and (34) yields

119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)


119863]minus120591minus120574minus119899119911

119911]minus1 =

Γ (])Γ (120591 + 120574 + 119899)

119911120591+120574+119899minus1

119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911minus120574minus119899

Γ (1 minus 120574 minus 119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(79)

Combining (79) with (78) and making some elementarysimplifications the asserted result (77) follows


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

= (Γ (120591) Γ (1 + ] minus 120591) sin (120573120587)

times sin [(] minus 120591 + 120573 minus 120579) 120587])

times (Γ (]) Γ (120591 minus 120574 minus 120579 minus 1) Γ (1 + 120574 + 120579)

times sin [(] minus 120591 + 120573) 120587] sin [(120573 minus 120579 minus 120574) 120587])minus1

sdotsin (120579120587)

sin [(120579 + 120574) 120587]

infin


Γ (] minus 120579 minus 119899) Γ (120579 + 119899)Γ (2 + ] minus 120591 + 120574 minus 119899) Γ (minus120574 + 119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(80)


Proof Upon first substituting 120583 997891rarr 120579 and ] 997891rarr 120574 inTheorem 12 and then setting

120572 = ] minus 120591 119906 (119911) = 119911120591minus120573

V (119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(81)

in which both 119906(119911) and V(119911) satisfy the conditions of Theo-rem 12 we have

119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911Γ (1 + ] minus 120591) sin (120573120587) sin (120579120587) sin [(] minus 120591 + 120573 minus 120579) 120587]sin [(] minus 120591 + 120573) 120587] sin [(120573 minus 120579 minus 120574) 120587] sin [(120579 + 120574) 120587]

sdot

infin


119863]minus120591+120574+1minus119899119911


Γ (2 + ] minus 120591 + 120574 minus 119899) Γ (minus120574 + 119899)

sdot 119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(82)

Now by using (33) and (34) we find that

119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

119863]minus120591+120574+1minus119899119911


=Γ (] minus 120579 minus 119899)

Γ (120591 minus 120574 minus 120579 minus 1)119911120591minus120574minus120579minus2

119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (120579 + 119899)

Γ (1 + 120579 + 120574)

times 119911120579+120574

Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(83)

Thus finally the result (80) follows by combining (83) and(82)

5 Corollaries and Consequences

We conclude this paper by presenting some special cases ofthe main results These special cases and consequences aregiven in the form of the following corollaries

Setting 119896 = 3 inTheorem 13 and using the fact that [12 p1496 Remark 7]

lim119887rarr0

11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)] = 120582Γ (119904)

(120582 gt 0)

(84)

we obtain the following corollary given recently by Srivastavaet al [19]

Corollary 19 Under the hypotheses ofTheorem 13 the follow-ing expansion formula holds true

Φ(120588112058811990113120590

112059011990213)

12058211205821199011205721205831120583119902120573

(119911 119904 119886)

=3Γ (120573) Γ (3120572)

Γ (120572) Γ (120573 + 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

(3120572)119899

(120573 + 120572)119899

119911119899

(119899 + 119886)119904

sdot 1198651[3120572 + 119899 1 + 120572 minus 120573 1 + 120572 minus 120573 120573 + 2120572 + 119899 minus1 1]

(85)

where 1198651denotes the first Appell function defined by [11 p 22]

1198651[119886 119887

1 119887

2 119888 119909

1 119909

2]

=

infin

sum11989811198982=0

(119886)1198981+1198982

(1198871)1198981

(1198872)1198982

(119888)1198981+1198982

1199091198981

1

1198981

1199091198982

2

1198982

(max 100381610038161003816100381611990911003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 lt 1)

(86)



Putting 119901 minus 1 = 119902 = 0 and setting 1205881= 1 and 120582

1= 120583 in

Theorem 15 reduces to the following expansion formula givenrecently by Srivastava et al [20]


Θ120582

120583(119911 119904 119886 119887)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(111)

120583120572+119888119899+120574120572+120573(119911

1 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(87)


119911 =1199111

2+1199111



Letting 119887 = 0 in Theorem 18 we deduce the followingexpansion formula obtained by Srivastava et al [19]


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886)

=Γ (120591) Γ (1 + ] minus 120591)

Γ (]) Γ (120591 minus 120574 minus 120579 minus 1)

sdot (sin120573120587 sin [(] minus 120591 + 120573 minus 120579) 120587] sin 120579120587)

times (Γ (1 + 120574 + 120579) sin [(] minus 120591 + 120573) 120587]

times sin [(120573 minus 120579 minus 120574) 120587] sin [(120579 + 120574) 120587])minus1

sdot

infin


Γ (]minus120579minus119899) Γ (120579+119899)Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120582

(119911 119904 119886)

Γ (2 + ] minus 120591 + 120574 minus 119899) Γ (minus120574 + 119899)(89)




References

[1] H M Srivastava and J Choi Series Associated with Zeta andRelated Functions Kluwer Academic Publishers London UK2001

[2] H M Srivastava ldquoSome formulas for the Bernoulli and Eulerpolynomials at rational argumentsrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 129 pp 77ndash84 2000

[3] H M Srivastava and J Choi ldquoZeta and q-Zeta Functions andAssociated Series and Integralsrdquo Zeta and q-Zeta Functions andAssociated Series and Integrals 2012

[4] S P Goyal and R K Laddha ldquoOn the generalized RiemanZeta function and the generalized Lambert transformrdquo GanitaSandesh vol 11 pp 99ndash108 1997

[5] S-D Lin and H M Srivastava ldquoSome families of the Hurwitz-Lerch Zeta functions and associated fractional derivative andother integral representationsrdquo Applied Mathematics and Com-putation vol 154 no 3 pp 725ndash733 2004

[6] M Garg K Jain and S L Kalla ldquoA further study of generalHurwitz-Lerch zeta functionrdquoAlgebras Groups andGeometriesvol 25 pp 311ndash319 2008

[7] H M Srivastava R K Saxena T K Pogany and R SaxenaldquoIntegral and computational representations of the extendedHurwitz-Lerch zeta functionrdquo Integral Transforms and SpecialFunctions vol 22 no 7 pp 487ndash506 2011

[8] H M Srivastava ldquoGenerating relations and other results asso-ciated with some families of the extended Hurwitz-Lerch Zetafunctionsrdquo SpringerPlus vol 2 no 1 article 67 pp 1ndash14 2013

[9] H M Srivastava ldquoSome generalizations and basic (or 119902-)extensions of the Bernoulli Euler and Genocchi polynomialsrdquoApplied Mathematics and Information Sciences vol 5 no 3 pp390ndash444 2011

[10] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK 1984

[11] H M Srivastava and P W Karlsson Multiple Gaussian Hyper-geometric Series Halsted Press Chichester UK 1985

[12] H M Srivastava ldquoA new family of the 120582-generalized Hurwitz-Lerch zeta functions with applicationsrdquo Applied Mathematics ampInformation Sciences vol 8 no 4 pp 1485ndash1500 2014

[13] R K Raina and P K Chhajed ldquoCertain results involving a classof functions associated with the Hurwitz zeta functionrdquo ActaMathematica Universitatis Comenianae vol 73 no 1 pp 89ndash100 2004

[14] H M Srivastava M J Luo and R K Raina ldquoNew resultsinvolving a class of generalized Hurwitz-Lerch zeta functionsand their applicationsrdquo Turkish Journal of Analysis and NumberTheory vol 1 pp 26ndash35 2013

[15] M A Chaudhry and S M Zubair On a Class of IncompleteGamma Function with Applications Chapman and HallCRCPress Company Washington DC USA 2001

[16] A MMathai R K Saxena and H J HauboldTheH-FunctionTheory and Applications Springer London UK 2010

[17] H M Srivastava K C Gupta and S P GoyalTheH-Functionsof One and Two Variables with Applications South AsianPublishers New Delhi India 1982

[18] H M Srivastava S Gaboury and A Bayad ldquoExpansionformulas for an extendedHurwitz-Lerch zeta function obtainedvia fractional calculusrdquo Advances in Difference Equations Inpress

[19] H M Srivastava S Gaboury and R Tremblay ldquoNew relationsinvolving an extended multiparameter Hurwitz-Lerch zetafunction with applicationsrdquo International Journal of Analysisvol 2014 Article ID 680850 14 pages 2014

[20] H M Srivastava S Gaboury and B J Fugere ldquoFurtherresultsinvolving a class of generalized Hurwitz-Lerch zeta func-tionsrdquo Submitted to Russian Journal of Mathematical Physics

[21] P L Gupta R C Gupta S-H Ong and H M Srivastava ldquoAclass of Hurwitz-Lerch Zeta distributions and their applicationsin reliabilityrdquo Applied Mathematics and Computation vol 196no 2 pp 521ndash531 2008


[22] H M Srivastava ldquoSome extensions of higher transcendentalfunctionsrdquo The Journal of the Indian Academy of Mathematicsvol 35 pp 165ndash176 2013

[23] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematical Studies ElsevierNorth-HollandScience Publishers London UK 2006

[24] A Erdelyi ldquoAn integral equation involving Legendre polynomi-alsrdquo SIAM Journal on Applied Mathematics vol 12 pp 15ndash301964

[25] J Liouville ldquoMemoire sur le calcul des differentielles a indicesquelconquesrdquo Journal de lrsquoEcole polytechnique vol 13 pp 71ndash162 1832

[26] M Riesz ldquoLrsquointegrale de Riemann-Liouville et le probleme deCauchyrdquo Acta Mathematica vol 81 pp 1ndash222 1949

[27] T J Osler ldquoFractional derivatives of a composite functionrdquoSIAM Journal on Mathematical Analysis vol 1 pp 288ndash2931970

[28] T J Osler ldquoLeibniz rule for the fractional derivatives andan application to infinite seriesrdquo SIAM Journal on AppliedMathematics vol 18 no 3 pp 658ndash674 1970

[29] T J Osler Leibniz rule the chain rule and taylorrsquos theorem forfractional derivatives [PhD thesis] New York University 1970

[30] T J Osler ldquoFractional derivatives and Leibniz rulerdquoThe Amer-ican Mathematical Monthly vol 78 no 6 pp 645ndash649 1971

[31] J-L Lavoie T J Osler and R Tremblay ldquoFundamental prop-erties of fractional derivatives via Pochhammer integralsrdquo inFractional Calculus and Its Applications B Ross Ed vol 457 ofLecture Notes in Mathematics pp 323ndash356 Springer New YorkNY USA 1975

[32] R TremblayUne contribution a la theorie de la derivee fraction-naire [PhD thesis] Laval University Quebec Canada 1974

[33] K S Miller and B Ross An Introduction of the FractionalCalculus and Fractional Differential Equations John Wiley andSons Singapore 1993

[34] S Gaboury and R Tremblay ldquoSummation formulas obtained bymeans of the generalized chain rule for fractional derivativesrdquoSubmitted to Journal of Complex Analysis

[35] T J Osler ldquoTaylorrsquos series generalized for fractional derivativesand applicationsrdquo SIAM Journal on Mathematical Analysis vol2 pp 37ndash48 1971

[36] G H Hardy ldquoRiemannrsquos forms of Taylorrsquos seriesrdquo Journal of theLondon Mathematical Society vol 20 no 1 pp 48ndash57 1945

[37] O Heaviside Electromagnetic Theory vol 2 Dover PublishingCompany New York NY USA 1950

[38] B Riemann Versuch einer Allgemeinen Auffasung der Inte-gration und Differentiation The Collected Works of BernhardRiemann Dover Publishing Company New York NY USA1953

[39] Y Watanabe ldquoZum Riemanschen binomischen Lehrsatzrdquo Pro-ceedings of the Physico-Mathematical Society of Japan vol 14 pp22ndash35 1932

[40] R Tremblay S Gaboury and B-J Fugere ldquoTaylor-like expan-sion in terms of a rational function obtained by means of frac-tional derivativesrdquo Integral Transforms and Special Functionsvol 24 no 1 pp 50ndash64 2013

[41] R Tremblay and B-J Fugere ldquoThe use of fractional derivativesto expand analytical functions in terms of quadratic functionswith applications to special functionsrdquo Applied Mathematicsand Computation vol 187 no 1 pp 507ndash529 2007

[42] R Tremblay S Gaboury and B-J Fugere ldquoA new Leibniz ruleand its integral analogue for fractional derivativesrdquo IntegralTransforms and Special Functions vol 24 no 2 pp 111ndash128 2013








Algebra




Geometry












Operations Research








Contents





















Toka Diagana










1 Introduction




119886(119899 119896) and 119878



(119905)119899=

119899

sum119896=0

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

(119905 minus 119886)119896

(1)

(119905 minus 119886)119899

=

119899

sum119896=0

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(119905)119896 (2)


119904119886(119899 119896) =

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

119878119886(119899 119896) =

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(3)


119904119886(119899 + 1 119896) = 119904

119886(119899 119896 minus 1) + (119886 minus 119899) 119904

119886(119899 119896) (4)

119878119886(119899 + 1 119896) = 119878

119886(119899 119896 minus 1) + (119896 minus 119886) 119878

119886(119899 119896) (5)


119904119886(0 0) = 1 119904

119886(119899 0) = (119886)

119899 119904

119886(0 119896) = 0 119899 119896 = 0

119878119886(0 0) = 1 119878

119886(119899 0) = (minus119886)

119899

119878119886(0 119896) = 0 119899 119896 = 0

(6)


1+ 119883


119896of





119875 [119882 = 119899] =119896

(1 minus 120579)minus119904

(minus log (1 minus 120579))119896

120579119899

119899(minus1)

119899minus119896

119904minus119904(119899 119896)

119875 [119885 = 119899] =119896

119890119898 (119890119897 minus 1)119896

119897119899

119899(minus1)

119899minus119896

119878minus119898119897

(119899 119896)

(7)






119870(1)

119899119903= 119903 + [log(119899 minus 1


1

119903+ 119900 (1)] (8)


2 Explicit Formula



119865119899(119909) = prod

119899


1198651015840

119899(119909) = sum


119899minus1le119899

119894isinN11989911989511198952119895119899minus1

1198911015840

119894(119909)

119899minus1

prod119896=1

119891119895119896(119909) (9)


1 2

119889

119889120585(120585119895 + 119886)

1= 119895

119889

119889120585(120585119895 + 119886)

2= (120585119895 + 119886) 119895 + (120585119895 + 119886 minus 1) 119895

(10)


1

119895119899119889

119889120585(120585119895 + 119886)



119899minus1

prod119896=1

(120585 +119886 minus 119895

119896

119895) (11)



1

119895119899119889119896

119889120585119896(120585119895 + 119886)


119899minus119896le119899minus1

119896

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(12)


1

119895119899119889119898

119889120585119898(120585119895 + 119886)


119899minus119898le119899minus1

119898

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(13)

Then

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899


119899minus119898le119899minus1

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(14)




119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895) = sum



prod119902=1

(120585 +119886 minus 119894

119902

119895)

(15)

119894119902

isin 1198951 1198952 119895

119899minus119898 has ( 119899minus119898





119902=1(120585 + (119886 minus





119895119899minus119898minus1





119902=1(120585 + (119886 minus 119895


of (1119895119899)(119889119898+1119889120585119898+1)(120585119895 + 119886)119899 Thus we have

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899



119898 + 1

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)



119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(16)

Lemma 2 Let 119904(119899 119896 119886) =

(1119896)lim120585rarr0

((sum119896

119895=0(minus1)

119896minus119895

( 119896119895) (119889119896119889120585119896)(120585119895 + 119886)

119899)119896)

Then


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (17)

Proof Using Lemma 1

119904 (119899 119896 119886)

=1

119896lim120585rarr0

((

119896

sum119895=0


(119896

119895) 119895119899


119899minus119896le119899minus1

119896

times

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)) (119896)

minus1

)

(18)



(120585 + (119886 minus 119895119902)119895) = (1119895119899minus119896)prod

119899minus119896

119902=1(120585119895 + 119886 minus 119895

119902)


1

119896lim120585rarr0

[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(120585119895 + 119886 minus 119895119902)]

]

=1

119896[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)]

]

(19)

which boils down to


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (20)

since

1

119896

119896

sum119895=0


(119896

119895) 119895119896

= 119878 (119896 119896) = 1 (21)



119904119886(119899 119896) = 119904 (119899 119896 119886) = sum


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (22)

Proof We know that


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (23)



119899+1minus119896)







119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (24)

is equal to


119899minus119896+1le119899minus1

119899minus119896+1

prod119902=1

(119886 minus 119895119902) + (119886 minus 119899)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)

(25)

This implies that

119904 (119899 + 1 119896 119886) = 119904 (119899 119896 minus 1 119886) + (119886 minus 119899) 119904 (119899 119896 119886) (26)



119886(119899 119896)


120582 = (1205821ge 120582


119896) (27)


119894


14= 11989115

= 11989123

= 11989131

= 11989142

= 1 119891119894119895= 0 elsewhere

(1 le 119895 le 120582119894)

0 0 0 1 10 0 11 0 00 10

(28)



(119886119894)119894ge0



1198860 1198861 119886

ℎ then |119879119889119860(ℎ 119903)| = ( ℎ+1




120596-tableau is











119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)


120596 (|119888|) (29)




is equal to 119904119886(119899 119896)



0 1 2 119899 minus 1 and

120596119860(120601) = prod

119888isin120601

120596 (|119888|)

=

119899minus119896

prod119894=1

(119886 minus 119895119894) 119895

119894isin 0 1 2 119899 minus 1

(30)

If 119886 = 1198861+ 1198862for some 119886

1and 119886

2 then with 120596lowast(119895) = 119886

2minus 119895

120596119860(120601) =

119899minus119896

prod119894=1

(1198861+ 120596lowast

(119895119894))

=

119899minus119896

sum119903=0

119886119899minus119896minus119903

1sum


119903le119895119899minus119896

119903

prod119894=1

120596lowast

(119902119894)

(31)






may write

120596119860(120601) = sum

120595isin119861120601

120596119860(120595) (32)


120596119860(120595) = 119886

119899minus119896minus119903

1

119903

prod119894=1

120596lowast

(119902119894) (33)

where 1199021 1199022 119902

119903isin 119895

1 1198952 119895


written as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

sum120595isin119861120601

120596119860(120595) (34)



1 1198952 119895

119899minus119896



(119899

119896)(

119899 minus 119896

119903) (35)





( 119899119896) ( 119899minus119896

119903)

( 119899119903)

= (119899 minus 119903

119896) (36)


119904119886(119899 119896) =

119899minus119896

sum119903=0

(119899 minus 119903

119896) 119886119899minus119896minus119903

1sum120595isin119861119903


120596lowast

(|119888|) (37)



119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

1sum



120596lowast

(|119888|) (38)

Clearly 119861119899minus119895



tity

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

11199041198862

(119899 119895) (39)


1and 119886

2





119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (40)






2isin 1198791198891198602(119895 minus 1 119895 minus 119899+ 119896)


2 we obtain an


119860

(119898 +


sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601)

=

119899

sum119896=0

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011)

times

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

(41)


Note that

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)


119895minus119899+119896le119898+119895minus1

119895minus119899+119896

prod119902=1

(119886 minus 119892119902)



119895minus119899+119896

prod119902=1

(119886 minus (119898 + 119892119902))

= 119904119886minus119898

(119895 119899 minus 119896)

(42)


sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011) = 119904

119886(119898 119896)

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601) = 119904

119886(119898 + 119895 119899)

(43)

Thus

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (44)



convolution formula

119904119886(119899 + 1119898 + 119895 + 1) =

119899

sum119896=0

119904119886(119896 119899) 119904

119886minus(119896+1)(119899 minus 119896 119895) (45)

Proof Let


in 1198601= 0 1 119896 minus 1


are in 1198602= 119896 + 1 119899








21199092 + sdot sdot sdot + 119886

119899119909119899 has only

real zeros then

1198862

119896ge 119886119896+1

119886119896minus1

119896

119896 minus 1

119899 minus 119896 + 1

119899 minus 119896(119896 = 2 119899 minus 1) (46)


sum119896=0

119904119886(119899 119896) (119905 + 119886)

119896

(47)



119896=0119904119886(119899 119896)119905119896 are 119886 119886minus1 119886minus



119899


and hence unimodal


⟨119905⟩119899=

119899

sum119896=0


119904119886(119899 119896) (119905 + 119886)

119896

(48)



119904119886(119899 119896) = (minus1)

119899minus119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902) (49)


119886(119899 119896)| as

1003816100381610038161003816119904119886 (119899 119896)1003816100381610038161003816 = (minus1)

119899minus119896

119904119886(119899 119896)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902)

(50)



lt 1199062


infin

sum119894=1

1

119906119894

= infin

infin

sum119894=1

1

1199062119894

lt infin (51)




which sum119899119896


119870119899= 119899 minus [

119899

sum119894=1

1

119906119894

minus

119899

sum119894=1

1

1199062119894

(1 +1

119906119894

)

minus1

+ 119900 (1)] (52)


1

1+1

2+ sdot sdot sdot +

1

119899= log 119899 + 120574 + 119900 (1) (53)



119886(119899 119896)|

119899

119896=0



119899


mation

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (54)




Denoting by 119896119886119899



1= 119887 + 0 119906

2= 119887 + 1 119906


119896119886119899

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894)2(1 +

1

119887 + 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894) (119887 + 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894 + 1+ 119900 (1)]

(55)


119899minus1

sum119894=0

1

119887 + 119894 + 1= log (119887 + 119899) minus log 119887 (56)

From this we get

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (57)



119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (58)



1)(119886 minus


119899minus119896)where 119895


1 By Theorem 10 119904119886(119899 119896) = sum

119899119899minus119896 Thus with 119906

1= 119886 119906

2=

119886 minus 1 119906119899minus1

= 119886 minus 119899 + 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894)2(1 +

1

119886 minus 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894) (119886 minus 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119886 minus 119894 + 1+ 119900 (1)]

(59)


119899119896 0 1 2 30 11 12 2 13 6 6 14 24 35 10 15 120 225 85 156 720 1624 735 1757 5040 13132 6769 19608 40320 109584 118124 672849 362880 1026576 1172700 72368010 3628800 10628640 12753576 8409500


119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (60)


9


119896minus19

= [log(1 + 8

1) + 119900 (1)]

= [log 9 + 119900 (1)]

asymp 2

(61)


10


1198961010

= [log( 10 + 1

10 minus 10 + 1) + 119900 (1)]

= [log 11 + 119900 (1)]

asymp 2

(62)







(63)


Table 2 Values of 11990410(119899 119896)

119899119896 0 1 2 30 11 10 12 90 19 13 720 242 27 14 5040 2414 431 345 30240 19524 5000 6356 151200 127860 44524 81757 604800 662640 305956 772248 1814400 2592720 1580508 5376289 3628800 6999840 5753736 265576410 3628800 10628640 12753576 8409500


119904(119899 119896) 0 1 2 3 4 50 11 0 12 0 1 13 0 2 3 14 0 6 11 6 15 0 24 50 35 10 16 0 120 274 225 85 157 0 720 1764 1624 735 1758 0 540 13068 13132 6769 19609 0 40320 109584 118124 67284 2244910 0 362880 1026576 1172700 723680 269325





2= 2 119906


119896119899= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

1

1198942(1 +

1

119894)minus1

+ 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

(1

119894minus

1

119894 + 1) + 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894 + 1+ 119900 (1)]

(64)


119899minus1

sum119894=1

1

1 + 119894= log 119899 minus log 1 + 120574 + 119900 (1) (65)

Therefore we have

119896119899= [log 119899 + 120574 + 119900 (1)] (66)



7

119896=0occurs at

1198967= [log 7 + 120574 + 119900 (1)]

= [195 + 05772 + 119900 (1)]

= [253 + 119900 (1)]

asymp 2

(67)


11989610

= [log 10 + 120574 + 119900 (1)]

= [230 + 05772 + 119900 (1)]

= [28772 + 119900 (1)]

asymp 3

(68)




Acknowledgment


References
















1 Introduction















2 New Instructions







x

f(x)

i

(ii) and (iii)1

c













2119888 and



















2minus119890


minus119873

2minus119873

] (1)









2119909 multiplica-








1sdot finite

1+ finite

2= 2 (2)








2120583 in the case






2((maximum



Inputsign exp mant

Drop leading zeros

numZeros + 1


Add

Switch

s e12 m51

lookup output


exp

Add 1

expP1

adjustedE

Switch

bitRot


Add 0x7ff

Add 0x3ff

Switch

Concatenate bit

xor

Add drop carry

moreThan22Zeros

isNot12

Concatenate bit

51

42

42 + 1

44

30

31

52

52

52

52


Add



(c)

(b)

(d)

(e)

(f)

(g)


(h)

(i)

(j)

(k)

leading12Mant

(a)

(p)

(r)

(o)

(l)

(m)

(n)

(q)

(s)

(t)

(u)

(v)

5

sim

sim

0x7ffSubs

Subs 0b0 0




12-bit lookup

12-bit lookup

ge1

e(last 10bits)lt1

sim


0b10 0Subs

Subs 0b0 0

low 10bits4

6

11

11

11

11

12

Lower 10






2119872

recip 3 113 376 0500 2048 3 infin






3 Evaluation





4 Conclusion




(a)


recip (2minus119890

119888)ext (2minus119890



119888)sat

sqrt (2minus119890

119888)ext 21198902

119888 0infin 0 NaN 0 0 0 NaN




(b)


minusinfin 119888 119888 0 0

infin 119888 0 119888 0

NaN 119888 119888 119888 0

(c)



minusinfin119891

minusinfin119891

infin infin119891

infin119891

infin119891

infin119891

NaN0

NaN119891

NaN119891

NaN119891

NaN119891

NaN1

2119891

2119891

2119891

2119891

NaN2

1radic2119891

1radic2119891

1radic2119891

1radic2119891

NaN3

0119891

0119891

0119891

0119891

recipdiv

sqrtrsqrtcbrt

rcbrtqdrt

rqdrtlog2

loglog1pexp2

expexpm1

atan2atanasinacos

sincostan

0 5 10 15 20 25 30







Acknowledgments


References

























1 Introduction








[119865 (119905)]119896

= sum119899gt0

119865 (119899 119896) 119905119899

(1)





119860Δ

(119899 119896) = 120572119896

119865Δ

(119899 119896) (2)



119860Δ

(119899 119896) = 120572119899

119865Δ

(119899 119896) (3)


119891(119899)119905119899 119877(119905) = sum119899gt0

119903(119899)119905119899


119860Δ

(119899 119896) =

119899

sum119898=119896

119865Δ

(119899119898)119866Δ

(119898 119896) (4)


119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899



119886(119899)119905119899

are

119886 (119899) =

119899

sum119896=1

119865Δ

(119899 119896) 119903 (119896)

119886 (0) = 119903 (0)

(5)




119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899


119899ge0119886(119899)119905119899 are

119886 (119899) =

119899

sum119896=0

119891 (119896) 119903 (119899 minus 119896) (6)




119910119899(119909) =

119899

sum119896=0

(119899 + 119896)

(119899 minus 119896)119896(119909

2)119896

(7)


119901119899(119909) = 119909

119899

119910119899minus1

(1

119909) (8)


119901119899(119909) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)(1

2)119899minus119896

119909119896 (9)


sum119899=0

119901119899(119909)

119899119905119899


(10)




119862Δ

(119899 119896) =119896

119899(2119899 minus 119896 minus 1

119899 minus 1) (11)



2)

119861 (119905) = 119890119905

=

infin

sum119899=0

1

119899119905119899

(12)


119860Δ

(119899 119896) = (2119909)119896

(1

2)119899

119862Δ

(119899 119896) = 2119896minus119899

119909119896119896

119899(2119899 minus 119896 minus 1

119899 minus 1)

(13)


119899=0119887(119899)119905119899

are equal to

119887 (119899) =1

119899 (14)


119901119899(119909) = 119899

119899

sum119896=1

119860Δ

(119899 119896) 119887 (119896) =

119899

sum119896=1

(2119899 minus 119896 minus 1)


119909119896

(15)




119898119899+1

(119909 120573 119888) =(119888 minus 1)

119899

119888119899

sdot ((119909 minus 119887119899)119898119899(119909 120573 119888) minus 120582

119899119898119899minus1

(119909 120573 119888))

(16)

where

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) = 119909 minus 119887

0

119887119899=

(1 + 119888) 119899 + 120573119888

1 minus 119888

120582119899=

119888119899 (119899 + 120573 minus 1)

(1 minus 119888)2

(17)


1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) =

119909 (119888 minus 1) + 120573119888

119888

1198982(119909 120573 119888)

=1199092 (119888 minus 1)

2

+ 119909 ((2120573 + 1) 1198882 minus 2120573119888 minus 1) + (120573 + 1) 1205731198882

1198882

(18)


infin

sum119899=0

119898119899(119909 120573 119888)

119899119905119899

= (1 minus119905

119888)119909


(19)





119860 (119905) = (1 minus119905

119888)119909

=

infin

sum119899=0

(119909

119899) (minus1)

119899

119888minus119899

119905119899


=

infin

sum119899=0


119899) (minus1)

119899

119905119899

(20)


suminfin

119899=0119886(119899)119905119899 and 119861(119905) = sum

infin


follows

119886 (119899) = (119909

119899) (minus1)

119899

119888minus119899

119887 (119899) = (minus119909 minus 120573

119899) (minus1)

119899

(21)


119898119899(119909 120573 119888) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896)

= (minus1)119899

119899

119899

sum119896=0

(119909

119896)(


119899 minus 119896) 119888minus119896

(22)



119872119899+1

(119909 120575 120578) = (119909 minus 119887119899)119872119899(119909 120575 120578) minus 120582

119899119872119899minus1

(119909 120575 120578)

(23)

where1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 119887

0

119887119899= (2119899 + 120578) 120575

120582119899= (1205752

+ 1) 119899 (119899 + 120578 minus 1)

(24)


1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 120575120578

1198722(119909 120575 120578) = 119909

2

minus 1199092120575 (1 + 120578) + 120578 ((120578 + 1) 1205752

minus 1)

(25)


infin

sum119899=0

119872119899(119909 120575 120578)

119899119905119899

= ((1 + 120575119905)2

+ 1199052

)minus1205782


1 + 120575119905))

(26)




119860 (119905) = ((1 + 120575119905)2

+ 1199052

)minus1205782

= (1 + 2120575119905 + (1205752

+ 1) 1199052

)minus1205782

119861 (119905) = exp (119909tanminus1 ( 119905

1 + 120575119905))

(27)


1(1198602(119905)) and we expand 119860


theorem

1198601(119905) = (1 + 119905)

minus1205782

=

infin

sum119899=0

(minus120578

2119899

) 119905119899

1198602(119905) = 2120575119905 + (120575

2

+ 1) 1199052

(28)


infin

119899=01198861(119899)119905119899

are

1198861(119899) = (

minus120578

2119899

) (29)


1198862119896minus119899

119887119899minus119896

(119896

119899 minus 119896) (30)


119860Δ

2(119899 119896) = (2120575)

2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896) (31)


infin

119899=0119886(119899)119905119899

119886 (0) = 1198861(0)

119886 (119899) =

119899

sum119896=1

119860Δ

2(119899 119896) 119886

1(119896)

(32)


119886 (0) = 1

119886 (119899) =

119899

sum119896=1

(2120575)2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896)(

minus120578

2119896

) (33)


1(1198612(119905)) where

1198611(119905) = 119890

119905

=

infin

sum119899=0

1

119899119905119899

1198612(119905) = 119909tanminus1 ( 119905

1 + 120575119905)

(34)



functions 1199091198621(1198622(119905)) where

1198621(119905) = tanminus1 (119905)

1198622(119905) =

119905

1 + 120575119905

(35)


((minus1)(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896] (36)

where [119895



119862Δ

1(119899 119896) = ((minus1)

(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

sdot

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896]

(37)


(119899 minus 1

119896 minus 1) 119886119896

119887119899minus119896

(38)


to

119862Δ

2(119899 119896) = (

119899 minus 1

119896 minus 1) (minus120575)

119899minus119896

(39)


2(119905)

119861Δ

2(119899 119896) = 119909

119896

119899

sum119898=119896

119862Δ

2(119899119898) 119862

Δ

1(119898 119896) (40)


infin

119899=01198871(119899)119905119899

are defined by

1198871(119899) =

1

119899 (41)


infin

119899=0119887(119899)119905119899

119887 (0) = 1198871(0)

119887 (119899) =

119899

sum119896=1

119861Δ

2(119899 119896) 119887

1(119896)

(42)


119887 (0) = 1

119887 (119899) =

119899

sum119896=1

119909119896

119899

119899

sum119898=119896

(119899

119898)120575119899minus119898

(1 + (minus1)

119898+119896

(minus1)(119898+119896)2minus119899

)

sdot

119898

sum119895=119896


(119895 minus 1)(119898

119895)[

119895

119896]

(43)


119872119899(119909 120575 120578) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896) (44)



Acknowledgment


References





































119892 (119911) =1

119911+ 1198870+

infin

sum119899=1

119887119899119911119899 (1)




1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911) (2)






119899| le

(119861 minus 119860)(119899 + 1) for 119892 isin Σ[119860 119861]



ℎ (119908) = 119892minus1

(119908) =1

119908+

infin

sum119899=0

119861119899119908119899

=1


1

119899119870119899

119899+1(1198870 1198871 119887

119899) 119908119899

(3)

where 119908 isin D

119870119899

119899+1(1198870 1198871 119887

119899)

= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)


3119887119899minus4

0(1198874+ 311988711198872)

+ sum119895ge5

119887119899minus119895

0119881119895

(4)


variables 1198871 1198872 119887

119899





119899|



119899| and




119899=1119901119899119911119899 be so that 119877119890(119901(119911)) gt 0


100381610038161003816100381610038161199012+ 1205721199012

1

10038161003816100381610038161003816le 2 + 120572

1003816100381610038161003816119901110038161003816100381610038162

(5)



119899=1119888119899119911119899


100381610038161003816100381610038161198882+ 1205741198882

1

10038161003816100381610038161003816le 1 + (120574 minus 1)

1003816100381610038161003816119888110038161003816100381610038162

(6)

Proof Write

119901 (119911) =[1 + 120593 (119911)]

[1 minus 120593 (119911)] (7)

where 119901(119911) = 1+suminfin



1= 21198881and 119901

2=

2(1198882+ 11988821)


2= 2(1198882+ 11988821) in (5) we

obtain100381610038161003816100381610038162 (1198882+ 1198882

1) + 120572 (2119888

1)210038161003816100381610038161003816le 2 + 120572

10038161003816100381610038162119888110038161003816100381610038162 (8)

or100381610038161003816100381610038161198882+ (1 + 2120572) 119888

2

1

10038161003816100381610038161003816le 1 + 2120572

1003816100381610038161003816119888110038161003816100381610038162

(9)




(119894)10038161003816100381610038161198870

1003816100381610038161003816 le

119861 minus 119860


119861 minus 119860 119900119905ℎ119890119903119908119894119904119890

(119894119894)10038161003816100381610038161198871

1003816100381610038161003816

le

119861 minus 119860


2 (119861 minus 119860)|1198870|2

119894119891 0 le 119860 le 1 minus 2119861

119861 minus 119860

2 119900119905ℎ119890119903119908119894119904119890

(119894119894119894)100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2

(119894V) 10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 119894119891 119887

119896= 0 119891119900119903 0 le 119896 le 119899 minus 1

(10)


1199111198921015840 (119911)

119892 (119911)= minus1 minus

infin

sum119899=0

119865119899+1

(1198870 1198871 1198872 119887

119899) 119911119899+1

(11)

where 119865119899+1

(1198870 1198871 1198872 119887




5=



119865119899+1

(1198870 1198871 119887

119899)


119899+1=119899+1

119860 (1198941 1198942 119894

119899+1) 1198871198941

01198871198942

1sdot sdot sdot 119887119894119899+1

119899

(12)

where

119860 (1198941 1198942 119894

119899+1)

= (minus1)(119899+1)+2119894

1+sdotsdotsdot+(119899+2)119894

119899+1

(1198941+ 1198942+ sdot sdot sdot + 119894

119899+1minus 1) (119899 + 1)

(1198941) (1198942) sdot sdot sdot (119894

119899+1)

(13)



119908ℎ1015840 (119908)


infin

sum119899=1

119865119899+1

(1198610 1198611 1198612 119861

119899) 119908119899+1

(14)

where119865119899+1

(1198610 1198611 1198612 119861


119899 + 1 given by

119865119899+1

= minus119899 (119899 minus (119899 + 1))

119899 (119899 minus 2119899)119861119899

0

minus119899 (119899 minus (119899 + 1))


01198611

minus119899 (119899 minus (119899 + 1))


01198612

minus119899 (119899 minus (119899 + 1))


0

sdot (1198613+119899 minus (2119899 minus 3)

21198612


119861119899minus119895

0119870119895

(15)


variables 1198611 1198612 119861

119899minus1and

1198610= minus1198870 119861

119899= minus

1

119899119870119899

119899+1(1198870 1198871 119887

119899) (16)


1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198881 1198882 119888

119899 119861) 119911119899

(17)

119908ℎ1015840 (119908)

ℎ (119908)= minus

1 + 119860120595 (119908)

1 + 119861120595 (119908)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198891 1198892 119889

119899 119861) 119908

119899

(18)


119899=1119888119899119911119899 and 120595(119908) = sum

infin

119899=1119889119899119908119899 are two



119899(1198961 1198962 119896

119899 119861) are given by

119870119901

119899(1198961 1198962 119896

119899 119861)

=119901

(119901 minus 119899)119899119896119899

1119861119899minus1

+119901


11198962119861119899minus2

+119901


11198963119861119899minus3

+119901


1

sdot [1198964119861119899minus4

+119901 minus 119899 + 3

21198962

2119861]

+119901


1

sdot [1198965119861119899minus5

+ (119901 minus 119899 + 4) 11989621198963119861]

+ sum119895ge6

119896119899minus119895

1119883119895

(19)


variables 1198962 1198963 119896

119899


minus119865119899+1

(1198870 1198871 1198872 119887

119899) = (119860 minus 119861)119870

minus1

119899+1(1198881 1198882 119888

119899+1 119861)

(20)


minus 119865119899+1

(1198610 1198611 1198612 119861

119899)

= (119860 minus 119861)119870minus1

119899+1(1198891 1198892 119889

119899+1 119861)

(21)


1198870= (119860 minus 119861) 119888

1

minus1198870= (119860 minus 119861) 119889

1

(22)

21198871minus 1198872

0= (119861 minus 119860) (119888

2

1119861 minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (119889

2

1119861 minus 1198892)

(23)




0and then adding

them gives

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus 1198611198882

1minus 1198611198892

1) (24)



21198872

0= (119861 minus 119860) (119888

2+ 1198892minus

2119861

(119861 minus 119860)21198872

0) (25)


sides we obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(119861 minus 119860)

2

(1198882+ 1198892)

2 (2119861 minus 119860)le

119861 minus 119860

radic2119861 minus 119860 (26)



119861 minus 119860



we obtain

41198871= (119861 minus 119860) [(119889

2minus 1198882) minus (119889

2

1minus 1198882

1) 119861] = (119861 minus 119860) (119889

2minus 1198882)

(28)


100381610038161003816100381611988711003816100381610038161003816 le

119861 minus 119860

2 (29)


21198871= (119861 minus 119860) (119889

2+ 1198601198892

1minus 2119861119889

2

1) (30)


100381610038161003816100381611988711003816100381610038161003816 le

1

2(119861 minus 119860) (

100381610038161003816100381610038161198892+ 1198601198892

1

10038161003816100381610038161003816+ 2119861

1003816100381610038161003816119889110038161003816100381610038162

)

le1

2(119861 minus 119860) (1 + (119860 minus 1)

1003816100381610038161003816119889110038161003816100381610038162

+ 211986110038161003816100381610038161198891

10038161003816100381610038162

)

=119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

(31)




2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

lt119861 minus 119860

2 (32)


21198871minus 1198872

0= (119861 minus 119860) (

119861

(119861 minus 119860)21198872

0minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (

119861

(119860 minus 119861)21198872

0minus 1198892)

(33)


100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2 (34)


(119899 + 1) 119887119899= (119860 minus 119861) 119888

119899+1 (35)


obtain

10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 (36)







0|2 given by






References



























1 Introductions


119891 (119911) = 119911 +

infin

sum120581=2

119886120581119911120581

(1)


119867[119886 119899] = 119901 isin 119867 119901 (119911) = 119886 + 119886119899119911119899

+119886119899+1

119911119899+1


(2)


119865 (119911) = 119911120573

+

infin

sum120581=2

120582120581119886120581119911120573+120581minus1

120573 isin N

120582120581= 120573120581minus1

119911 =1

120573 120573 |119911| lt 1

(3)


(119891 lowast 119892) (119911) =

infin

sum119896=0

119886119896119887119896119911119896

(4)


119896=0119886119896119911119896 and 119892(119911) = sum

infin

119896=0119887119896119911119896 are analytic in

119880

Let 120595(119911) = 119911120573 + sum

infin



120595 (119911) lowast119911120573

1 minus 120573119911 119911 =

1

120573 120573 isin N 120573 |119911| lt 1 (5)



119876(120485) (120572 120591 120574 120573)

= 119865 (119911) isin 119867 (120573) R119890

times120572[119865(119911)]

(120485)

[119911120573](120485)

+120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

gt 120574 119911 isin 119880

(6)

where

119911 = 0 120573 gt 120485 120485 isin N cup 0 also 120572 + 120591 = 120574 (7)







119891 (119911) =1

120572 + 120591int|119909|


times

infin

sum120581=0

(120572 + 120591) 119909120581

119911120581+1

(120581 + 1) 120591 + 120572] 119889120583 (119909)

(8)




Θ120572120573

(119911) = (120573int119911

0

119905120573minus120572minus1

[119891 (119905)]120572

119892 (119905) 119889119905)

1120573

(10)






119901 (119911) = int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909) (|119911| lt 1) (11)



119892 (119911) +1199111198921015840 (119911)

119888≺ ℎ (119911) (12)

then

119892 (119911) ≺ 119902 (119911) =119888119911minus119888120581

120581int119911

0

119905(119888minus120581)120581

ℎ (119905) 119889119905 ≺ ℎ (119911) (13)



1199111199011015840 (119911)

119901 (119911)≺ ℎ (119911) (14)

then

119901 (119911) ≺ 119902 (119911) = 119886 exp [1

119899int119911

0

ℎ (120585) 120585minus1

119889120585] (15)




(ii) |1199011015840(119911)| le 2R119890119901(119911)(1 minus 1199032)


100381610038161003816100381610038161003816100381610038161003816

1199111199011015840 (119911)

119901 (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 119901 (119911)

(1 minus 119903)2

(16)




(120485)

(120572120591120574120573)




119911120485

[119865 (119911)](120485)

=

120573

119875120485

(120572 + 120591)int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120573 minus 120485) (120572 + 120591) 119909120581119911120573+120581

(120572 + 120591) (120573 minus 120485) + 120581120591] 119889120583 (119909)

(17)

where 119899119875119903




119901 (119911)

=

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

isin 119867 [1 119899]

(18)

ByTheorem 3

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

= int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909)

(19)


[119865 (119911)](120485+1)

[119911120573](120485+1)

+120572[119865 (119911)]

(120485)

120591[119911120573](120485)

=1

120591int|119909|=1

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909119911

1 minus 119909119911119889120583 (119909)

(20)

which yields

119911((120572+120591)(120485minus120573)+120591)120591

int119911

0

[[119865 (120585)]

(120485+1)

[120585120573](120485+1)

+120572[119865 (120585)]

(120485)

120591[120585120573](120485)

]


119889120585

=1

120591int|119909|=1

119911((120572+120591)(120485minus120573)+120591)120591

times int119911

0

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909120585

1 minus 119909120585


119889120585 119889120583 (119909)

(21)



shows that 119865(119911) isin 119876(120485)(120572 120591 120574 120573)



119865 (119911) =1

120572 + 120591int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581119911120573+120581

120572120573 + 120591 (120581 + 120573)] 119889120583 (119909)

(22)


1




10038161003816100381610038161198861205811003816100381610038161003816 le 119867 (120572 120591 120574 120573) (23)

where

119867(120572 120591 120574 120573) =2 (120572 + 120591 minus 120574) (120573 minus 120485) [

120573

119875120485]

120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(24)


119911120485

[119865119909(119911)](120485)

120573119875120485

= 119911120573

+

infin

sum120581=2


(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)(|119909| = 1)

(25)




119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [[120581 minus 1] 119903

120581minus1]

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

le2119903R119890 Ψ (119911)

(1 minus 119903)2

(26)


119866 (119911) equiv120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

+120572[119865(119911)]

(120485)

[119911120573](120485)

minus 120574 (27)

and then119911Ψ1015840 (119911)

Ψ (119911)=

suminfin

120581=2120599 (120572 120591 120574 120573) [120581 minus 1] 119886

120581119911120581minus1

1 + suminfin

120581=2120599 (120572 120591 120574 120573) 119886

120581119911120581minus1

(28)

where

120599 (120572 120591 120574 120573) =120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591 minus 120574) [120573

119875(120485)

]

(29)

From (23) and (28) we got100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [120581 minus 1] 119903

120581minus1

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

(120581 ge 2) (30)


119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 Ψ (119911)

(1 minus 119903)2

(31)

Sinceinfin

sum120581=2


lt 2119903R119890 Ψ (119911)

1 +

infin

sum120581=2

2 (120573 minus 120485) 119903120581minus1

gt (1 minus 119903)2

(32)




(120485)

(120572 120591 120574 120573)



1199111198661015840 (119911)

119866 (119911) ≺ ℎ (119911) (33)

then

119866 (119911) ≺ 119902 (119911) = (120572 + 120591 minus 120574) exp [119899minus1

int119911

0

ℎ (120585) 120585minus1

119889120585 ] (34)



119866 (119911) = (120572 + 120591 minus 120574) +

infin

sum120581=2

120582120581[ [120573+120581minus1]119875

120485+1] 119886120581119911120581minus1

120573119875120485+1

(35)


1199111198661015840

(119911)

119866 (119911) ≺ ℎ (119911) (36)




1199111198661015840 (119911)

119866 (119911) ≺

radic119911

1 minus 119911(37)

then

119866 (119911) ≺ (120572 + 120591 minus 120574) 1 + radic119911

1 minus radic119911

1119899

120572 + 120591 = 120574 (38)




120573 gt 120485 + 1 with 120572 + 120591 = 0 If

119866 (119911) ≺ (120572 + 120591)1 + 119860119911

1 + 119861119911 (39)

then[119865(119911)]

(120485)

[119911120573](120485)

≺(120572 + 120591) (120573 minus 120485)

120591120581119911(120572+120591)(120485minus120573)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(40)


1

(120572 + 120591)119866 (119911) = 1 +

infin

sum120581=2

119862120581119886120581119911120581minus1

(41)

where

119862120581=

120582120581[ (120573+120581minus1)119875

120485+1] sdot [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591) [120573

119875120485+1

] (42)

Let 119892(119911) = [119865(119911)](120485)

[119911120573](120485) then 119892(119911) = 1 + sum

infin

120581=2120603120581119886120581119911120581minus1

where 120603120581= 120582120581[(120573+120581minus1)

119875120485+1

][(120573)

119875120485] and from (39) we have

119892 (119911) +120591

(120572 + 120591) (120573 minus 120485)1199111198921015840

(119911) ≺ (1 + 119860119911

1 + 119861119911) = ℎ (119911) (43)


119892 (119911) ≺(120572 + 120591) (120573 minus 120485)

120591120581119911minus(120572+120591)(120573minus120485)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(44)


Corollary 16 Let

119876(0)

(120572 120591 0 120573) =120572 (119865 (119911))

119911120573+

120591[119865 (119911)]1015840

[119911120573]1015840

(45)

If

119876(0)

(120572 120591 0 120573) ≺ (120572 + 120591) (1 + 119860119911

1 + 119861119911) (46)

then

119865 (119911)

119911120573≺

120572120573 + 120591 (120573 minus 120581)

120591120581119911minus120573(120572+120591)120591120581

times int119911

0

120585(120573(120572+120591)minus120591120581)120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(47)






Acknowledgments



References





















1 Introduction


119891 (119911) =119892 (119911)

ℎ (119911) (1)






R ℎ (119911) gt 0 (119911 isin 119880lowast

) (2)


119891 ≺ 119892 in 119880 or 119891 (119911) ≺ 119892 (119911) (119911 isin 119880lowast

) (3)


|119908 (119911)| le |119911| 119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880lowast

) (4)


119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0)

119891 (119880) sube 119892 (119880) (119911 isin 119880lowast

) (5)





2 Preliminaries


119891 (119911) =1

119911+

infin

sum119899=1

119886119899119911119899

(6)




119891119895(119911) =

1

119911+

infin

sum119899=1

119886119899119895119911119899

(7)


and 1198912(119911) by

(1198911lowast 1198912) =

1

119911+

infin

sum119899=1

11988611989911198861198992119911119899

(8)


119902120582120583

(119911) =1

119911+

infin

sum119899=1

(120582

119899 + 1 + 120582)

120583

119911119899

(120582 gt 0 120583 ge 0) (9)



119871120582120583119891 (119911) = (119891 (119911) lowast 119902

120582120583(119911))

=1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(10)


119905119886119892 and 119871


tively by

119871120582120583119892 (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198871198991003816100381610038161003816 119911119899


119871120582120583ℎ (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198881198991003816100381610038161003816 119911119899


ℎ (119911) isin Σ)

(11)


119871120582120583

(119891 lowast 119892) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast)

119871120582120583

(119891 lowast ℎ) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198881198991003816100381610038161003816 119911119899


)

(12)






property

120588119871120582120583

(119891 lowast 119892) (119911)

119871120582120583

(119891 lowast ℎ) (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911 (119911 isin 119880

lowast

) (13)


and 119871(120582 120583)(119891 lowast ℎ)(119911) = 0


120601 (120572 120573 119911) =1

119911+

infin

sum119899=0

(120572)119899+1

(120573)119899+1

119911119899

(14)



120601 (120572 120573 119911) =1

119911 21198651(1 120572 120573 119911) (15)

where

21198651(119887 120572 120573 119911) =

infin

sum119899=0

(119887)119899(120572)119899

(120573)119899

119911119899

119899(16)


120582120583(119911)


119871 (120572 120573 120582 120583) 119891 (119911) = (119891 (119911) lowast 120601 (120572 120573 119911) lowast 119902120582120583

(119911))

=1

119911+

infin

sum119899=1

(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(17)




119911(119871 (120572 120573 120582 120583) 119891 (119911))1015840

= 120572119871 (120572 + 1 120573 120582 120583) 119891 (119911)

minus (120572 + 1) 119871 (120572 120573 120582 120583) 119891 (119911) (18)


120588119871 (120572 + 1 120573 120582 120583) 119891 (119911)

119871 (120572 120573 120582 120583) 119891 (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911

(0 le 119861 lt 119860 le 1 120588 gt 0)

(19)



relation (19) above






infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861)

minus10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(20)


119891119899(119911) =

1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

(21)



1003816100381610038161003816100381610038161003816100381610038161003816

120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899) 119911119899+1

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899119911119899+1

)

minus11003816100381610038161003816100381610038161003816100381610038161003816

le 120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899)10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899

10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816)

minus1

le 1

(22)





100381610038161003816100381610038161003816100381610038161003816lt 1

(119911 isin 119880lowast)

(23)


Σ120583




120582(120588 119860 119861) then

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119899 ge 1

(24)


(21)


120582(120588 119860 119861)



1

119903minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903

le1003816100381610038161003816119891 (119911)

1003816100381610038161003816

le1

119903+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903


1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

le100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(25)



inequalityinfin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(26)


1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

le1

119903+ 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

119903+ 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|minus

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

ge1

119903minus 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

ge1

119903minus 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(27)


sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(119860 minus 119861) (2 + 120582)

120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(28)

Hence

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2+

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

le1

1199032+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

ge1

1199032minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816

ge1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(29)



120582(120588 119860 119861) which




disk |119911| lt 1199031 where

1199031= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(30)



119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (31)

For |119911| lt 1199031 we have

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 1 + 120582))

120583

119886119899119911119899

1119911 + suminfin

119899=1(120582(119899 + 1 + 120582))

120583

119886119899119911119899

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

100381610038161003816100381610038161003816100381610038161003816

lesuminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(32)


sum119899=1

(119899 + 1) (120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1


infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

)

(33)

or

suminfin

119899=1(119899 + 2 minus 120575) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(1 minus 120575)le 1 (34)


(119899 + 2 minus 120575) (120582(119899 + 1 + 120582))120583

|119911|119899+1

(1 minus 120575)

le (120582

119899 + 120582 + 1)

120583

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899 ge 1

(35)



|119911| le (1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899+1

(36)




disk |119911| lt 1199032 where

1199032= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(37)



1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891 (119911))10158401015840

(119891 (119911))1015840+ 2

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (38)






infin

sum119899=1

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(39)


119891119899(119911)

=1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

119899 ge 1

(40)




120582(120588 120572 120573 119860 119861) then1003816100381610038161003816119886119899

1003816100381610038161003816

le(119860 minus 119861) (119899 + 120582 + 1)

120583

(120573)119899+1

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861)) (120572)

119899+1

119899 ge 1

(41)


(40)




1199033= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(42)





1199034= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(43)






Acknowledgment


References






















1 Introduction


Φ (119911 119904 119886) =

infin

sum119899=0

119911119899

(119899 + 119886)119904


0 119904 isin C when |119911| lt 1

R (119904) gt 1 when |119911| = 1)

(1)


119904(120585) defined by

120577 (119904) =

infin

sum119899=1

1

119899119904= Φ (1 119904 1) = 120577 (119904 1) (R (119904) gt 1)

120577 (119904 119886) =

infin

sum119899=0

1

(119899 + 119886)119904


0)

ℓ119904(120585) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 1)119904= Φ (e2120587119894120585 119904 1) (R (119904) gt 1 120585 isin R)

(2)


119904(119911)

Li119904(119911) =

infin

sum119899=1

119911119899

119899119904= 119911Φ (119911 119904 1)


(3)


120601 (120585 119886 119904) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 119886)119904= Φ (e2120587119894120585 119904 119886)



when 120585 isin Z)

(4)




Φ (119911 119904 119886) =1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

1 minus 119911eminus119905d119905

(R (119886) gt 0R (119904) gt 0 when |119911| ≦ 1 (119911 = 1) R (119904) gt 1

when 119911 = 1) (5)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) =

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

119911119899

(119899 + 119886)119904

(119901 119902 isin N0 120582

119895isin C (119895 = 1 119901)

119886 120583119895isin C Z

minus

0(119895 = 1 119902)

120588119895 120590

119896isin R

+

(119895 = 1 119901 119896 = 1 119902)





lowast

)

(6)

with

nablalowast

= (

119901

prod119895=1

120588minus120588119895

119895) sdot (

119902

prod119895=1

120590120590119895

119895)

Δ =

119902

sum119895=1

120590119895minus

119901

sum119895=1

120588119895

Ξ = 119904 +

119902

sum119895=1

120583119895minus

119901

sum119895=1

120582119895+119901 minus 119902

2

(7)



(120582)120581=Γ (120582 + 120581)

Γ (120582)

=


1 (120581 = 0 120582 isin C 0)

(8)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0)

(9)



119898or

119897Ψ119898(119897 119898 isin N

0) involved


119897119865119898(119897 119898 isin N




1198871 119887

119898such that

119886119895isin C (119895 = 1 119897) 119887

119895isin C Z

minus

0(119895 = 1 119898)

(10)


119897Ψlowast

119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

=

infin

sum119899=0

(1198861)1198601119899sdot sdot sdot (119886

119897)119860119897119899

(1198871)1198611119899sdot sdot sdot (119887

119898)119861119898119899

119911119899

119899

=Γ (119887


119898)

Γ (1198861) sdot sdot sdot Γ (119886

119897) 119897Ψ119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

(119860119895gt 0 (119895 = 1 119897) 119861

119895gt 0 (119895 = 1 119898)

1 +

119898

sum119895=1

119861119895minus

119897

sum119895=1

119860119895≧ 0)

(11)


|119911| lt nabla = (

119897

prod119895=1

119860minus119860119895

119895) sdot (

119898

prod119895=1

119861119861119895

119895) (12)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

Γ (119904)intinfin

0


119905120582)

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0)

(13)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 0 120582)

= Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

= e119887Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 0)

(14)


119901 minus 1 = 119902 = 0 (1205821= 120583 120588

1= 1) (15)


Θ120582

120583(119911 119904 119886 119887)

=1

Γ (119904)intinfin

0


119905120582) (1 minus 119911eminus119905)

minus120583

d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0 120583 isin C)

(16)


R (119904) gt 0 when 119887 = 0 |119911| ≦ 1 (119911 = 1) (17)

or

R (119904 minus 120583) gt 0 when 119887 = 0 119911 = 1 (18)




Θ120582


Θ120582

120583(119911 119904 119886 0) = Φ

lowast

120583(119911 119904 119886) =

1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

(1 minus 119911eminus119905)120583d119905

(R (119886) gt 0 R (119904) gt 0 when |119911| ≦ 1 (119911 = 1)

R (119904 minus 120583) gt 0 when 119911 = 1)

(19)



[4] as follows

Φlowast

120583(119911 119904 119886) =

infin

sum119899=0

(120583)119899

(119886 + 119899)119904

119911119899

119899 (20)


120583(119911 119904 119886 119887) that is


Θ1

1(1 119904 119886 119887)

= 120577119887(119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1


119905) d119905

(21)




(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899

119899(120582 gt 0)

(22)



119867119898119899

pq (119911)

= 119867119898119899

pq [119911 |(119886

1 119860

1) (119886p 119860p)

(1198871 119861

1) (119887q 119861q)

]

=1

2120587119894intL

Ξ (119904) 119911minus119904d119904 (119911 isin C 0

1003816100381610038161003816arg (119911)1003816100381610038161003816 lt 120587)

(23)

where

Ξ (119904) =prod

119898

119895=1Γ (119887

119895+ 119861

119895119904) sdot prod

119899

119895=1Γ (1 minus 119886

119895minus 119860

119895119904)

prodp

119895=119899+1Γ (119886

119895+ 119860

119895119904) sdot prod

q

119895=119898+1Γ (1 minus 119887

119895minus 119861

119895119904)

(24)


119895gt 0 (119895 =

1 p) 119861119895gt 0 (119895 = 1 q) 119886

119895isin C (119895 = 1 p)



Γ (119887119895+ 119861

119895119904)

119898

119895=1

(25)


Γ (1 minus 119886119895+ 119860

119895119904)

119899

119895=1

(26)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902





1 minus 119905)

=

infin

sum119899=0

119871(120572)

119899(119909) 119905

119899

(|119905| lt 1 120572 isin C)

(27)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=eminus119887

Γ (119904)

infin

sum119899=0

119899

sum119896=0

(minus1)119896

(119899

119896) sdot Γ (119904 + 120582 (120572 + 119896 + 1))

sdot 119871(120572)

119899(119887)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 + 120582 (120572 + 119895 + 1) 119886)

(R (119886) gt 0 R (119904 + 120582120572) gt minus120582)

(28)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) (29)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



D120572

119911119911

119901

119891 (119911) =1

Γ (minus120572)int119911

0

119891 (120585) 120585119901


(R (120572) lt 0 R (119901) gt 1)

(30)


D120572

119911119911

119901

119891 (119911) =d119898

d119911119898D

120572minus119898

119911119911

119901

119891 (119911) (31)



Im(120585)

Branch line for

Branch line for


exp[p(ln(g(120585)))]

C1

C2

C3

C4

Re(120585)

z

a

gminus1(0)





119901


119863120572

119892(119911)[119892 (119911)]

119901

119891 (119911)

=eminus119894120587119901Γ (1 + 120572)4120587 sin (120587119901)

times int119862(119911+119892

minus1(0)+119911minus119892

minus1(0)minus119865(119886)119865(119886))

sdot119891 (120585) [119892 (120585)]

119901

1198921015840 (120585)

[119892 (120585) minus 119892 (119911)]120572+1

d120585

(32)



1cup119862

2cup119862

3cup119862








119892(119911)[119892(119911)]

119901



119863120572

119911119911

119901

=Γ (1 + 119901)

Γ (1 + 119901 minus 120572)119911119901minus120572

(R (119901) gt minus1) (33)



11205881199011205901120590119902)

12058211205821199011205831120583119902


119863120572

119911119911

120573minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899)

1198871120582

| (119904 1) (01

120582)]119863

120572

119911119911

120573minus1119911119899

119899

=Γ (120573)

120582Γ (119904) Γ (120573 minus 120572)

infin

sum119899=0

(120573)119899

(120573 minus 120572)119899

sdot

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899+120573minus120572minus1

119899

=Γ (120573)


Φ(1205881120588119901112059011205901199021)

12058211205821199011205731205831120583119902120573minus120572

(119911 119904 119886 119887 120582)







119863120572

119892(119911)119891 (119911) = 119863

120572

ℎ(119911)119891 (119911) 119892

1015840(119911)

ℎ1015840(119911)(ℎ(119908) minus ℎ(119911)

119892(119908) minus 119892(119911))

120572+1

100381610038161003816100381610038161003816100381610038161003816119908=119911

(35)




119911119901119874120572

120573119891 (119911)

= 119901(119911119901minus1

)minus120572

sdot119911

119874120572

120573119891 (119911) (119911

119901minus1

)120572

119901minus1

prod119904=1

(1 minus119911

119908eminus(2120587119894119904)119901)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(36)

where

119892(119911)119874120572


Γ (120573)

Γ (120572)[119892 (119911)]

1minus120573

119863120572minus120573

119892(119911)[119892 (119911)]

120572minus1

sdot sdot sdot

(37)



120579 (119911) = (119911 minus 1199110) 119902 (119911) (38)


119870

=0 1 [119888] ([119888] 119905ℎ119890 119897119886119903119892119890119904119905 119894119899119905119890119892119890119903 119899119900119905 119892119903119890119886119905119890119903 119905ℎ119886119899 119888)

(39)


0and let

120596 = exp (2120587119894119886) (40)



sum119896isin119870

119888minus1

120596minus120574119896

119891 (120579minus1

(120579 (119911) 120596119896

))

=

infin


[120579 (119911)]119888119899+120574

Γ (119888119899 + 120574 + 1)

sdot119863119888119899+120574

119911minus119887119891 (119911) 120579

1015840

(119911) (119911minus119911

0

120579 (119911))

119888119899+120574+1

100381610038161003816100381610038161003816100381610038161003816119911=1199110

(1003816100381610038161003816119911minus1199110

1003816100381610038161003816=100381610038161003816100381611991101003816100381610038161003816)

(41)



119891 (119911) = 119888

infin


(119911 minus 1199110)119888119899+120574

Γ(119888119899 + 120574 + 1)119863

119888119899+120574

119911minus119887119891(119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199110

(42)



1)(119911minus119911

2)) where 119911

1and 119911

2are two



120596 = exp(2120587119894119886) (43)


1and 119911


curves

119862 (119905) 119862 (119905) subR 0 lt 119905 ≦ 119903 (44)

be defined by

119862 (119905) = 1198621(119905) cup 119862

2(119905)

= 119911 1003816100381610038161003816120582119905 (1199111 1199112 119911)

1003816100381610038161003816 =1003816100381610038161003816100381610038161003816120582119905(119911

1 119911

21199111+ 119911

2

2)1003816100381610038161003816100381610038161003816

(45)

where

120582119905(119911

1 119911

2 119911) = [119911 minus

1199111+ 119911

2

2+ 119905 (

1199111minus 119911

2

2)]

sdot [119911 minus (1199111+ 119911

2

2) minus 119905 (

1199111minus 119911

2

2)]

(46)


1+ 119911



1199111+ 119911

2

2+ (

1199111minus 119911

2

2) 119905 (47)


2

2minus (

1199111minus 119911

2

2) 119905 (48)

Im(120585)

Branch line for

Branch line for

C1

C2

z1

z2

Re(120585)

z

c =z1 + z2

2

(120585 minus z2)120583120579(120585)

aminus120574minus1

(120585 minus z1)120579(120585)

aminus120574minus1



[(119911 minus 1199111) (119911 minus 119911

2)]

120582


2))))

(49)


plusmn

defined by

119871plusmn=

119911 119911 =1199111+ 119911

2

2plusmn 119905 (

1199111minus 119911

2

2) (0 ≦ 119905 ≦ 1)

119911 119911 =1199111+ 119911

2

2plusmn 119894119905 (

1199111minus 119911

2

2) (119905 lt 0)

(50)



1)(119911minus119911

2) gt 0 (v)


2)119901


+cup 119871


119911minus1199112

(119911 minus 1199112)119901


119870 = 119896 119896 isin N arg(120582119905(119911

1 119911

21199111+ 119911

2

2))

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) +

2120587119896

119886

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) + 2120587

(51)


defined by

120585 =1199111+ 119911

2

2+1199111minus 119911

2


sum119896isin119870

119888minus1120596minus120574119896

1199111minus 119911

2

119891 (120601minus1

(120596119896

120601 (119911)))

times [120601minus1

(120596119896

120601(119911)) minus 1199111]][120601

minus1

(120596119896

120601(119911)) minus 1199112]120583

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

[120601 (119911)]119888119899+120574

(52)


where

120601 (119911) =119911 minus 119911

1

119911 minus 1199112

(53)



2)120583

1199111minus 119911

2

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

(119911 minus 119911

1

119911 minus 1199112

)

119888119899+120574

(54)


2) where 119911

1and 119911




sum119896isin119870

119888minus1

120596minus120574119896

[(1199112minus 119911

1+ radicΔ

119896

2)

120572

(1199111minus 119911

2+ radicΔ

119896

2)

120573

sdot 119891 (1199111+ 119911

2+ radicΔ

119896

2)


sdot (1199112minus 119911

1minus radicΔ

119896

2)

120572


2minus radicΔ

119896

2)

120573

119891(1199111+ 119911

2minus radicΔ

119896

2)]

=

infin



eminus119894120587119888(119899+1)[120579 (119911)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911minus1199112


(120579 (119911)

(119911minus1199112) (119911minus119911

1))


1205791015840

(119911)119891 (119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199111

(55)

where

Δ119896= (119911

1minus 119911

2)2

+ 4119881 (120596119896

120579 (119911))

119881 (119911) =

infin

sum119903=1

119863119903minus1

119911[119902 (119911)]

minus119903

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

119911119903

119903

120579 (119911) = (119911 minus 1199111) (119911 minus 119911

2) 119902 ((119911 minus 119911

1) (119911 minus 119911

2))

(56)


(119911 minus 1199111)(119911 minus 119911

2)) and 119911


119891 (119911) = 119888119911minus120573

(119911 minus 1199111)minus120572

times

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot119863minus120572+119888119899+120574

119911119911


(119911 + 119908 minus 1199111)119891(119911)

10038161003816100381610038161003816119911=1199111

(119908=119911)

(57)



119863120572

119911119911

119901+119902

119906 (119911) V (119911)

=

infin


(120572

120574 + 119899)119863

120572minus120574minus119899

119911119911

119901

119906 (119911)119863120574+119899

119911119911

119902V (119911) (58)




119863120572

119911119911

120572+120573minus1

119906 (119911) V (119911)


sdot

infin


119863120572+]+1minus119899119911



(60)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 (1120582))]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot 119865(119896minus1)



where 120582 gt 0 and 119865(119899)119863


119865(119899)

119863[119886 119887

1 119887

119899 119888 119909

1 119909

119899]

=

infin

sum1198981119898119899=0

(119886)1198981+sdotsdotsdot+119898

119899

(1198871)1198981

sdot sdot sdot (119887119899)119898119899

(119888)1198981+sdotsdotsdot+119898

119899

1199091198981

1


119909119898119899

119899

119898119899

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091198991003816100381610038161003816 lt 1)

(62)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119896

(119911119896minus1)120572

sdot119911119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) (119911119896minus1

)120572

sdot

119896minus1

prod119904=1

(1 minus119911

119908eminus2120587119894119904119896)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(63)




119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

(64)



119896

(119911119896minus1)120572

119911

119874120572

120573

times

1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119911119899+(119896minus1)120572

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


sdot((119911119908) eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119908=119911

=119896

120582Γ (119904) (119911119896minus1)120572

times

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981



sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

times119911

119874120572

120573119911


119896minus1+119899+(119896minus1)120572

=119896

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


times(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

sdotΓ (120573) Γ (119898


119896minus1+ 119899 + 119896120572)

Γ (120572) Γ (120573 + 1198981+ sdot sdot sdot + 119898

119896minus1+ 119899 + 119896120572)


=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 1120582)]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(119899+119896120572)1198981+sdotsdotsdot+119898

119896minus1

(1+120572minus120573)1198981



119896minus1

sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

(65)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


119911minus1198881198990(119911 minus 119911

0)119888119899

Γ (119888119899 + 1) Γ (1 minus 119888119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(1003816100381610038161003816119911 minus 1199110

1003816100381610038161003816 =100381610038161003816100381611991101003816100381610038161003816 120582 gt 0)

(66)


Proof Setting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


0 we

have

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


(119911 minus 1199110)119888119899

Γ (1 + 119888119899)

sdot119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

(67)


0| = |119911

0|


119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

=119911minus1198881198990

Γ (1 minus 119888119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(68)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(69)


119911 =1199111

2+1199111



Proof By taking 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 9 with 119911

2= 0 120583 = 120572 ] = 120573 and 0 lt 119888 ≦ 1 we find

that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888(119911 minus 1199111)minus120573

119911minus120572

1199111

sdot

infin



sdot (119911 minus 119911

1

119911)119888119899+120574

sdot119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

(71)



119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120572+120573minus1

1

Γ (120572 + 119888119899 + 120574)

Γ (120572 + 120573)

times Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582)

(72)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573+120574

(119911 minus 1199111)minus120572+120574

119911120573+120572minus2120574minus1

1

sdot

infin


sin [(120573 minus 119888119899 minus 120574) 120587] e119894120587119888(119899+1)

sin [(120573 + 119888 minus 120574) 120587] Γ (1 minus 120572 + 119888119899 + 120574)

sdot (119911(119911 minus 119911

1)

11991121

)

119888119899

Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdot [(119911 minus 1199111)Φ

(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (120573 minus 119888119899 minus 120574

120572 + 120573 minus 2119888119899 minus 2120574) 119911

1

sdotΦ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)]

(73)


119911 =1199111

2+1199111



Proof Putting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 10 with 119911

2= 0 0 lt 119888 ≦ 1 119902(119911) = 1 and 120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2) we find that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573

(119911 minus 1199111)minus120572

sdot

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

(75)


119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

= 119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

+ (119911 minus 1199111)119863

minus120572+119888119899+120574

119911


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120573+120572minus2119888119899minus2120574

1


Γ (1 + 120573 + 120572 minus 2119888119899 minus 2120574)

sdot Φ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (119911 minus 119911

1

1199111

)Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdotΦ(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582) )

(76)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

=Γ (120591) Γ (1 + ] minus 120591) sin (120574120587)

120587

sdot

infin


(minus1)119899

Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)


(77)



12058211205821199011205831120583119902


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=

infin


(] minus 120591120574 + 119899


119911]minus1

sdot 119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(78)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)



119911]minus1 =

Γ (])Γ (120591 + 120574 + 119899)

119911120591+120574+119899minus1

119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911minus120574minus119899

Γ (1 minus 120574 minus 119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(79)



Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

= (Γ (120591) Γ (1 + ] minus 120591) sin (120573120587)




sdotsin (120579120587)

sin [(120579 + 120574) 120587]

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(80)



120572 = ] minus 120591 119906 (119911) = 119911120591minus120573

V (119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(81)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


sdot

infin


119863]minus120591+120574+1minus119899119911




119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(82)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

119863]minus120591+120574+1minus119899119911


=Γ (] minus 120579 minus 119899)


119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (120579 + 119899)

Γ (1 + 120579 + 120574)

times 119911120579+120574

Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(83)





lim119887rarr0

11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)] = 120582Γ (119904)

(120582 gt 0)

(84)



Φ(120588112058811990113120590

112059011990213)

12058211205821199011205721205831120583119902120573

(119911 119904 119886)

=3Γ (120573) Γ (3120572)

Γ (120572) Γ (120573 + 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

(3120572)119899

(120573 + 120572)119899

119911119899

(119899 + 119886)119904


(85)


1198651[119886 119887

1 119887

2 119888 119909

1 119909

2]

=

infin

sum11989811198982=0

(119886)1198981+1198982

(1198871)1198981

(1198872)1198982

(119888)1198981+1198982

1199091198981

1

1198981

1199091198982

2

1198982

(max 100381610038161003816100381611990911003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 lt 1)

(86)




1= 120583 in



Θ120582

120583(119911 119904 119886 119887)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(111)

120583120572+119888119899+120574120572+120573(119911

1 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(87)


119911 =1199111

2+1199111





Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886)

=Γ (120591) Γ (1 + ] minus 120591)



times (Γ (1 + 120574 + 120579) sin [(] minus 120591 + 120573) 120587]


sdot

infin


Γ (]minus120579minus119899) Γ (120579+119899)Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120582

(119911 119904 119886)





References


















































Algebra




Geometry












Operations Research








Contents





















Toka Diagana










1 Introduction




119886(119899 119896) and 119878



(119905)119899=

119899

sum119896=0

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

(119905 minus 119886)119896

(1)

(119905 minus 119886)119899

=

119899

sum119896=0

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(119905)119896 (2)


119904119886(119899 119896) =

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

119878119886(119899 119896) =

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(3)


119904119886(119899 + 1 119896) = 119904

119886(119899 119896 minus 1) + (119886 minus 119899) 119904

119886(119899 119896) (4)

119878119886(119899 + 1 119896) = 119878

119886(119899 119896 minus 1) + (119896 minus 119886) 119878

119886(119899 119896) (5)


119904119886(0 0) = 1 119904

119886(119899 0) = (119886)

119899 119904

119886(0 119896) = 0 119899 119896 = 0

119878119886(0 0) = 1 119878

119886(119899 0) = (minus119886)

119899

119878119886(0 119896) = 0 119899 119896 = 0

(6)


1+ 119883


119896of





119875 [119882 = 119899] =119896

(1 minus 120579)minus119904

(minus log (1 minus 120579))119896

120579119899

119899(minus1)

119899minus119896

119904minus119904(119899 119896)

119875 [119885 = 119899] =119896

119890119898 (119890119897 minus 1)119896

119897119899

119899(minus1)

119899minus119896

119878minus119898119897

(119899 119896)

(7)






119870(1)

119899119903= 119903 + [log(119899 minus 1


1

119903+ 119900 (1)] (8)


2 Explicit Formula



119865119899(119909) = prod

119899


1198651015840

119899(119909) = sum


119899minus1le119899

119894isinN11989911989511198952119895119899minus1

1198911015840

119894(119909)

119899minus1

prod119896=1

119891119895119896(119909) (9)


1 2

119889

119889120585(120585119895 + 119886)

1= 119895

119889

119889120585(120585119895 + 119886)

2= (120585119895 + 119886) 119895 + (120585119895 + 119886 minus 1) 119895

(10)


1

119895119899119889

119889120585(120585119895 + 119886)



119899minus1

prod119896=1

(120585 +119886 minus 119895

119896

119895) (11)



1

119895119899119889119896

119889120585119896(120585119895 + 119886)


119899minus119896le119899minus1

119896

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(12)


1

119895119899119889119898

119889120585119898(120585119895 + 119886)


119899minus119898le119899minus1

119898

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(13)

Then

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899


119899minus119898le119899minus1

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(14)




119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895) = sum



prod119902=1

(120585 +119886 minus 119894

119902

119895)

(15)

119894119902

isin 1198951 1198952 119895

119899minus119898 has ( 119899minus119898





119902=1(120585 + (119886 minus





119895119899minus119898minus1





119902=1(120585 + (119886 minus 119895


of (1119895119899)(119889119898+1119889120585119898+1)(120585119895 + 119886)119899 Thus we have

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899



119898 + 1

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)



119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(16)

Lemma 2 Let 119904(119899 119896 119886) =

(1119896)lim120585rarr0

((sum119896

119895=0(minus1)

119896minus119895

( 119896119895) (119889119896119889120585119896)(120585119895 + 119886)

119899)119896)

Then


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (17)

Proof Using Lemma 1

119904 (119899 119896 119886)

=1

119896lim120585rarr0

((

119896

sum119895=0


(119896

119895) 119895119899


119899minus119896le119899minus1

119896

times

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)) (119896)

minus1

)

(18)



(120585 + (119886 minus 119895119902)119895) = (1119895119899minus119896)prod

119899minus119896

119902=1(120585119895 + 119886 minus 119895

119902)


1

119896lim120585rarr0

[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(120585119895 + 119886 minus 119895119902)]

]

=1

119896[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)]

]

(19)

which boils down to


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (20)

since

1

119896

119896

sum119895=0


(119896

119895) 119895119896

= 119878 (119896 119896) = 1 (21)



119904119886(119899 119896) = 119904 (119899 119896 119886) = sum


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (22)

Proof We know that


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (23)



119899+1minus119896)







119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (24)

is equal to


119899minus119896+1le119899minus1

119899minus119896+1

prod119902=1

(119886 minus 119895119902) + (119886 minus 119899)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)

(25)

This implies that

119904 (119899 + 1 119896 119886) = 119904 (119899 119896 minus 1 119886) + (119886 minus 119899) 119904 (119899 119896 119886) (26)



119886(119899 119896)


120582 = (1205821ge 120582


119896) (27)


119894


14= 11989115

= 11989123

= 11989131

= 11989142

= 1 119891119894119895= 0 elsewhere

(1 le 119895 le 120582119894)

0 0 0 1 10 0 11 0 00 10

(28)



(119886119894)119894ge0



1198860 1198861 119886

ℎ then |119879119889119860(ℎ 119903)| = ( ℎ+1




120596-tableau is











119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)


120596 (|119888|) (29)




is equal to 119904119886(119899 119896)



0 1 2 119899 minus 1 and

120596119860(120601) = prod

119888isin120601

120596 (|119888|)

=

119899minus119896

prod119894=1

(119886 minus 119895119894) 119895

119894isin 0 1 2 119899 minus 1

(30)

If 119886 = 1198861+ 1198862for some 119886

1and 119886

2 then with 120596lowast(119895) = 119886

2minus 119895

120596119860(120601) =

119899minus119896

prod119894=1

(1198861+ 120596lowast

(119895119894))

=

119899minus119896

sum119903=0

119886119899minus119896minus119903

1sum


119903le119895119899minus119896

119903

prod119894=1

120596lowast

(119902119894)

(31)






may write

120596119860(120601) = sum

120595isin119861120601

120596119860(120595) (32)


120596119860(120595) = 119886

119899minus119896minus119903

1

119903

prod119894=1

120596lowast

(119902119894) (33)

where 1199021 1199022 119902

119903isin 119895

1 1198952 119895


written as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

sum120595isin119861120601

120596119860(120595) (34)



1 1198952 119895

119899minus119896



(119899

119896)(

119899 minus 119896

119903) (35)





( 119899119896) ( 119899minus119896

119903)

( 119899119903)

= (119899 minus 119903

119896) (36)


119904119886(119899 119896) =

119899minus119896

sum119903=0

(119899 minus 119903

119896) 119886119899minus119896minus119903

1sum120595isin119861119903


120596lowast

(|119888|) (37)



119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

1sum



120596lowast

(|119888|) (38)

Clearly 119861119899minus119895



tity

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

11199041198862

(119899 119895) (39)


1and 119886

2





119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (40)






2isin 1198791198891198602(119895 minus 1 119895 minus 119899+ 119896)


2 we obtain an


119860

(119898 +


sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601)

=

119899

sum119896=0

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011)

times

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

(41)


Note that

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)


119895minus119899+119896le119898+119895minus1

119895minus119899+119896

prod119902=1

(119886 minus 119892119902)



119895minus119899+119896

prod119902=1

(119886 minus (119898 + 119892119902))

= 119904119886minus119898

(119895 119899 minus 119896)

(42)


sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011) = 119904

119886(119898 119896)

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601) = 119904

119886(119898 + 119895 119899)

(43)

Thus

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (44)



convolution formula

119904119886(119899 + 1119898 + 119895 + 1) =

119899

sum119896=0

119904119886(119896 119899) 119904

119886minus(119896+1)(119899 minus 119896 119895) (45)

Proof Let


in 1198601= 0 1 119896 minus 1


are in 1198602= 119896 + 1 119899








21199092 + sdot sdot sdot + 119886

119899119909119899 has only

real zeros then

1198862

119896ge 119886119896+1

119886119896minus1

119896

119896 minus 1

119899 minus 119896 + 1

119899 minus 119896(119896 = 2 119899 minus 1) (46)


sum119896=0

119904119886(119899 119896) (119905 + 119886)

119896

(47)



119896=0119904119886(119899 119896)119905119896 are 119886 119886minus1 119886minus



119899


and hence unimodal


⟨119905⟩119899=

119899

sum119896=0


119904119886(119899 119896) (119905 + 119886)

119896

(48)



119904119886(119899 119896) = (minus1)

119899minus119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902) (49)


119886(119899 119896)| as

1003816100381610038161003816119904119886 (119899 119896)1003816100381610038161003816 = (minus1)

119899minus119896

119904119886(119899 119896)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902)

(50)



lt 1199062


infin

sum119894=1

1

119906119894

= infin

infin

sum119894=1

1

1199062119894

lt infin (51)




which sum119899119896


119870119899= 119899 minus [

119899

sum119894=1

1

119906119894

minus

119899

sum119894=1

1

1199062119894

(1 +1

119906119894

)

minus1

+ 119900 (1)] (52)


1

1+1

2+ sdot sdot sdot +

1

119899= log 119899 + 120574 + 119900 (1) (53)



119886(119899 119896)|

119899

119896=0



119899


mation

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (54)




Denoting by 119896119886119899



1= 119887 + 0 119906

2= 119887 + 1 119906


119896119886119899

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894)2(1 +

1

119887 + 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894) (119887 + 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894 + 1+ 119900 (1)]

(55)


119899minus1

sum119894=0

1

119887 + 119894 + 1= log (119887 + 119899) minus log 119887 (56)

From this we get

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (57)



119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (58)



1)(119886 minus


119899minus119896)where 119895


1 By Theorem 10 119904119886(119899 119896) = sum

119899119899minus119896 Thus with 119906

1= 119886 119906

2=

119886 minus 1 119906119899minus1

= 119886 minus 119899 + 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894)2(1 +

1

119886 minus 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894) (119886 minus 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119886 minus 119894 + 1+ 119900 (1)]

(59)


119899119896 0 1 2 30 11 12 2 13 6 6 14 24 35 10 15 120 225 85 156 720 1624 735 1757 5040 13132 6769 19608 40320 109584 118124 672849 362880 1026576 1172700 72368010 3628800 10628640 12753576 8409500


119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (60)


9


119896minus19

= [log(1 + 8

1) + 119900 (1)]

= [log 9 + 119900 (1)]

asymp 2

(61)


10


1198961010

= [log( 10 + 1

10 minus 10 + 1) + 119900 (1)]

= [log 11 + 119900 (1)]

asymp 2

(62)







(63)


Table 2 Values of 11990410(119899 119896)

119899119896 0 1 2 30 11 10 12 90 19 13 720 242 27 14 5040 2414 431 345 30240 19524 5000 6356 151200 127860 44524 81757 604800 662640 305956 772248 1814400 2592720 1580508 5376289 3628800 6999840 5753736 265576410 3628800 10628640 12753576 8409500


119904(119899 119896) 0 1 2 3 4 50 11 0 12 0 1 13 0 2 3 14 0 6 11 6 15 0 24 50 35 10 16 0 120 274 225 85 157 0 720 1764 1624 735 1758 0 540 13068 13132 6769 19609 0 40320 109584 118124 67284 2244910 0 362880 1026576 1172700 723680 269325





2= 2 119906


119896119899= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

1

1198942(1 +

1

119894)minus1

+ 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

(1

119894minus

1

119894 + 1) + 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894 + 1+ 119900 (1)]

(64)


119899minus1

sum119894=1

1

1 + 119894= log 119899 minus log 1 + 120574 + 119900 (1) (65)

Therefore we have

119896119899= [log 119899 + 120574 + 119900 (1)] (66)



7

119896=0occurs at

1198967= [log 7 + 120574 + 119900 (1)]

= [195 + 05772 + 119900 (1)]

= [253 + 119900 (1)]

asymp 2

(67)


11989610

= [log 10 + 120574 + 119900 (1)]

= [230 + 05772 + 119900 (1)]

= [28772 + 119900 (1)]

asymp 3

(68)




Acknowledgment


References
















1 Introduction















2 New Instructions







x

f(x)

i

(ii) and (iii)1

c













2119888 and



















2minus119890


minus119873

2minus119873

] (1)









2119909 multiplica-








1sdot finite

1+ finite

2= 2 (2)








2120583 in the case






2((maximum



Inputsign exp mant

Drop leading zeros

numZeros + 1


Add

Switch

s e12 m51

lookup output


exp

Add 1

expP1

adjustedE

Switch

bitRot


Add 0x7ff

Add 0x3ff

Switch

Concatenate bit

xor

Add drop carry

moreThan22Zeros

isNot12

Concatenate bit

51

42

42 + 1

44

30

31

52

52

52

52


Add



(c)

(b)

(d)

(e)

(f)

(g)


(h)

(i)

(j)

(k)

leading12Mant

(a)

(p)

(r)

(o)

(l)

(m)

(n)

(q)

(s)

(t)

(u)

(v)

5

sim

sim

0x7ffSubs

Subs 0b0 0




12-bit lookup

12-bit lookup

ge1

e(last 10bits)lt1

sim


0b10 0Subs

Subs 0b0 0

low 10bits4

6

11

11

11

11

12

Lower 10






2119872

recip 3 113 376 0500 2048 3 infin






3 Evaluation





4 Conclusion




(a)


recip (2minus119890

119888)ext (2minus119890



119888)sat

sqrt (2minus119890

119888)ext 21198902

119888 0infin 0 NaN 0 0 0 NaN




(b)


minusinfin 119888 119888 0 0

infin 119888 0 119888 0

NaN 119888 119888 119888 0

(c)



minusinfin119891

minusinfin119891

infin infin119891

infin119891

infin119891

infin119891

NaN0

NaN119891

NaN119891

NaN119891

NaN119891

NaN1

2119891

2119891

2119891

2119891

NaN2

1radic2119891

1radic2119891

1radic2119891

1radic2119891

NaN3

0119891

0119891

0119891

0119891

recipdiv

sqrtrsqrtcbrt

rcbrtqdrt

rqdrtlog2

loglog1pexp2

expexpm1

atan2atanasinacos

sincostan

0 5 10 15 20 25 30







Acknowledgments


References

























1 Introduction








[119865 (119905)]119896

= sum119899gt0

119865 (119899 119896) 119905119899

(1)





119860Δ

(119899 119896) = 120572119896

119865Δ

(119899 119896) (2)



119860Δ

(119899 119896) = 120572119899

119865Δ

(119899 119896) (3)


119891(119899)119905119899 119877(119905) = sum119899gt0

119903(119899)119905119899


119860Δ

(119899 119896) =

119899

sum119898=119896

119865Δ

(119899119898)119866Δ

(119898 119896) (4)


119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899



119886(119899)119905119899

are

119886 (119899) =

119899

sum119896=1

119865Δ

(119899 119896) 119903 (119896)

119886 (0) = 119903 (0)

(5)




119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899


119899ge0119886(119899)119905119899 are

119886 (119899) =

119899

sum119896=0

119891 (119896) 119903 (119899 minus 119896) (6)




119910119899(119909) =

119899

sum119896=0

(119899 + 119896)

(119899 minus 119896)119896(119909

2)119896

(7)


119901119899(119909) = 119909

119899

119910119899minus1

(1

119909) (8)


119901119899(119909) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)(1

2)119899minus119896

119909119896 (9)


sum119899=0

119901119899(119909)

119899119905119899


(10)




119862Δ

(119899 119896) =119896

119899(2119899 minus 119896 minus 1

119899 minus 1) (11)



2)

119861 (119905) = 119890119905

=

infin

sum119899=0

1

119899119905119899

(12)


119860Δ

(119899 119896) = (2119909)119896

(1

2)119899

119862Δ

(119899 119896) = 2119896minus119899

119909119896119896

119899(2119899 minus 119896 minus 1

119899 minus 1)

(13)


119899=0119887(119899)119905119899

are equal to

119887 (119899) =1

119899 (14)


119901119899(119909) = 119899

119899

sum119896=1

119860Δ

(119899 119896) 119887 (119896) =

119899

sum119896=1

(2119899 minus 119896 minus 1)


119909119896

(15)




119898119899+1

(119909 120573 119888) =(119888 minus 1)

119899

119888119899

sdot ((119909 minus 119887119899)119898119899(119909 120573 119888) minus 120582

119899119898119899minus1

(119909 120573 119888))

(16)

where

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) = 119909 minus 119887

0

119887119899=

(1 + 119888) 119899 + 120573119888

1 minus 119888

120582119899=

119888119899 (119899 + 120573 minus 1)

(1 minus 119888)2

(17)


1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) =

119909 (119888 minus 1) + 120573119888

119888

1198982(119909 120573 119888)

=1199092 (119888 minus 1)

2

+ 119909 ((2120573 + 1) 1198882 minus 2120573119888 minus 1) + (120573 + 1) 1205731198882

1198882

(18)


infin

sum119899=0

119898119899(119909 120573 119888)

119899119905119899

= (1 minus119905

119888)119909


(19)





119860 (119905) = (1 minus119905

119888)119909

=

infin

sum119899=0

(119909

119899) (minus1)

119899

119888minus119899

119905119899


=

infin

sum119899=0


119899) (minus1)

119899

119905119899

(20)


suminfin

119899=0119886(119899)119905119899 and 119861(119905) = sum

infin


follows

119886 (119899) = (119909

119899) (minus1)

119899

119888minus119899

119887 (119899) = (minus119909 minus 120573

119899) (minus1)

119899

(21)


119898119899(119909 120573 119888) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896)

= (minus1)119899

119899

119899

sum119896=0

(119909

119896)(


119899 minus 119896) 119888minus119896

(22)



119872119899+1

(119909 120575 120578) = (119909 minus 119887119899)119872119899(119909 120575 120578) minus 120582

119899119872119899minus1

(119909 120575 120578)

(23)

where1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 119887

0

119887119899= (2119899 + 120578) 120575

120582119899= (1205752

+ 1) 119899 (119899 + 120578 minus 1)

(24)


1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 120575120578

1198722(119909 120575 120578) = 119909

2

minus 1199092120575 (1 + 120578) + 120578 ((120578 + 1) 1205752

minus 1)

(25)


infin

sum119899=0

119872119899(119909 120575 120578)

119899119905119899

= ((1 + 120575119905)2

+ 1199052

)minus1205782


1 + 120575119905))

(26)




119860 (119905) = ((1 + 120575119905)2

+ 1199052

)minus1205782

= (1 + 2120575119905 + (1205752

+ 1) 1199052

)minus1205782

119861 (119905) = exp (119909tanminus1 ( 119905

1 + 120575119905))

(27)


1(1198602(119905)) and we expand 119860


theorem

1198601(119905) = (1 + 119905)

minus1205782

=

infin

sum119899=0

(minus120578

2119899

) 119905119899

1198602(119905) = 2120575119905 + (120575

2

+ 1) 1199052

(28)


infin

119899=01198861(119899)119905119899

are

1198861(119899) = (

minus120578

2119899

) (29)


1198862119896minus119899

119887119899minus119896

(119896

119899 minus 119896) (30)


119860Δ

2(119899 119896) = (2120575)

2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896) (31)


infin

119899=0119886(119899)119905119899

119886 (0) = 1198861(0)

119886 (119899) =

119899

sum119896=1

119860Δ

2(119899 119896) 119886

1(119896)

(32)


119886 (0) = 1

119886 (119899) =

119899

sum119896=1

(2120575)2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896)(

minus120578

2119896

) (33)


1(1198612(119905)) where

1198611(119905) = 119890

119905

=

infin

sum119899=0

1

119899119905119899

1198612(119905) = 119909tanminus1 ( 119905

1 + 120575119905)

(34)



functions 1199091198621(1198622(119905)) where

1198621(119905) = tanminus1 (119905)

1198622(119905) =

119905

1 + 120575119905

(35)


((minus1)(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896] (36)

where [119895



119862Δ

1(119899 119896) = ((minus1)

(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

sdot

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896]

(37)


(119899 minus 1

119896 minus 1) 119886119896

119887119899minus119896

(38)


to

119862Δ

2(119899 119896) = (

119899 minus 1

119896 minus 1) (minus120575)

119899minus119896

(39)


2(119905)

119861Δ

2(119899 119896) = 119909

119896

119899

sum119898=119896

119862Δ

2(119899119898) 119862

Δ

1(119898 119896) (40)


infin

119899=01198871(119899)119905119899

are defined by

1198871(119899) =

1

119899 (41)


infin

119899=0119887(119899)119905119899

119887 (0) = 1198871(0)

119887 (119899) =

119899

sum119896=1

119861Δ

2(119899 119896) 119887

1(119896)

(42)


119887 (0) = 1

119887 (119899) =

119899

sum119896=1

119909119896

119899

119899

sum119898=119896

(119899

119898)120575119899minus119898

(1 + (minus1)

119898+119896

(minus1)(119898+119896)2minus119899

)

sdot

119898

sum119895=119896


(119895 minus 1)(119898

119895)[

119895

119896]

(43)


119872119899(119909 120575 120578) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896) (44)



Acknowledgment


References





































119892 (119911) =1

119911+ 1198870+

infin

sum119899=1

119887119899119911119899 (1)




1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911) (2)






119899| le

(119861 minus 119860)(119899 + 1) for 119892 isin Σ[119860 119861]



ℎ (119908) = 119892minus1

(119908) =1

119908+

infin

sum119899=0

119861119899119908119899

=1


1

119899119870119899

119899+1(1198870 1198871 119887

119899) 119908119899

(3)

where 119908 isin D

119870119899

119899+1(1198870 1198871 119887

119899)

= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)


3119887119899minus4

0(1198874+ 311988711198872)

+ sum119895ge5

119887119899minus119895

0119881119895

(4)


variables 1198871 1198872 119887

119899





119899|



119899| and




119899=1119901119899119911119899 be so that 119877119890(119901(119911)) gt 0


100381610038161003816100381610038161199012+ 1205721199012

1

10038161003816100381610038161003816le 2 + 120572

1003816100381610038161003816119901110038161003816100381610038162

(5)



119899=1119888119899119911119899


100381610038161003816100381610038161198882+ 1205741198882

1

10038161003816100381610038161003816le 1 + (120574 minus 1)

1003816100381610038161003816119888110038161003816100381610038162

(6)

Proof Write

119901 (119911) =[1 + 120593 (119911)]

[1 minus 120593 (119911)] (7)

where 119901(119911) = 1+suminfin



1= 21198881and 119901

2=

2(1198882+ 11988821)


2= 2(1198882+ 11988821) in (5) we

obtain100381610038161003816100381610038162 (1198882+ 1198882

1) + 120572 (2119888

1)210038161003816100381610038161003816le 2 + 120572

10038161003816100381610038162119888110038161003816100381610038162 (8)

or100381610038161003816100381610038161198882+ (1 + 2120572) 119888

2

1

10038161003816100381610038161003816le 1 + 2120572

1003816100381610038161003816119888110038161003816100381610038162

(9)




(119894)10038161003816100381610038161198870

1003816100381610038161003816 le

119861 minus 119860


119861 minus 119860 119900119905ℎ119890119903119908119894119904119890

(119894119894)10038161003816100381610038161198871

1003816100381610038161003816

le

119861 minus 119860


2 (119861 minus 119860)|1198870|2

119894119891 0 le 119860 le 1 minus 2119861

119861 minus 119860

2 119900119905ℎ119890119903119908119894119904119890

(119894119894119894)100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2

(119894V) 10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 119894119891 119887

119896= 0 119891119900119903 0 le 119896 le 119899 minus 1

(10)


1199111198921015840 (119911)

119892 (119911)= minus1 minus

infin

sum119899=0

119865119899+1

(1198870 1198871 1198872 119887

119899) 119911119899+1

(11)

where 119865119899+1

(1198870 1198871 1198872 119887




5=



119865119899+1

(1198870 1198871 119887

119899)


119899+1=119899+1

119860 (1198941 1198942 119894

119899+1) 1198871198941

01198871198942

1sdot sdot sdot 119887119894119899+1

119899

(12)

where

119860 (1198941 1198942 119894

119899+1)

= (minus1)(119899+1)+2119894

1+sdotsdotsdot+(119899+2)119894

119899+1

(1198941+ 1198942+ sdot sdot sdot + 119894

119899+1minus 1) (119899 + 1)

(1198941) (1198942) sdot sdot sdot (119894

119899+1)

(13)



119908ℎ1015840 (119908)


infin

sum119899=1

119865119899+1

(1198610 1198611 1198612 119861

119899) 119908119899+1

(14)

where119865119899+1

(1198610 1198611 1198612 119861


119899 + 1 given by

119865119899+1

= minus119899 (119899 minus (119899 + 1))

119899 (119899 minus 2119899)119861119899

0

minus119899 (119899 minus (119899 + 1))


01198611

minus119899 (119899 minus (119899 + 1))


01198612

minus119899 (119899 minus (119899 + 1))


0

sdot (1198613+119899 minus (2119899 minus 3)

21198612


119861119899minus119895

0119870119895

(15)


variables 1198611 1198612 119861

119899minus1and

1198610= minus1198870 119861

119899= minus

1

119899119870119899

119899+1(1198870 1198871 119887

119899) (16)


1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198881 1198882 119888

119899 119861) 119911119899

(17)

119908ℎ1015840 (119908)

ℎ (119908)= minus

1 + 119860120595 (119908)

1 + 119861120595 (119908)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198891 1198892 119889

119899 119861) 119908

119899

(18)


119899=1119888119899119911119899 and 120595(119908) = sum

infin

119899=1119889119899119908119899 are two



119899(1198961 1198962 119896

119899 119861) are given by

119870119901

119899(1198961 1198962 119896

119899 119861)

=119901

(119901 minus 119899)119899119896119899

1119861119899minus1

+119901


11198962119861119899minus2

+119901


11198963119861119899minus3

+119901


1

sdot [1198964119861119899minus4

+119901 minus 119899 + 3

21198962

2119861]

+119901


1

sdot [1198965119861119899minus5

+ (119901 minus 119899 + 4) 11989621198963119861]

+ sum119895ge6

119896119899minus119895

1119883119895

(19)


variables 1198962 1198963 119896

119899


minus119865119899+1

(1198870 1198871 1198872 119887

119899) = (119860 minus 119861)119870

minus1

119899+1(1198881 1198882 119888

119899+1 119861)

(20)


minus 119865119899+1

(1198610 1198611 1198612 119861

119899)

= (119860 minus 119861)119870minus1

119899+1(1198891 1198892 119889

119899+1 119861)

(21)


1198870= (119860 minus 119861) 119888

1

minus1198870= (119860 minus 119861) 119889

1

(22)

21198871minus 1198872

0= (119861 minus 119860) (119888

2

1119861 minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (119889

2

1119861 minus 1198892)

(23)




0and then adding

them gives

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus 1198611198882

1minus 1198611198892

1) (24)



21198872

0= (119861 minus 119860) (119888

2+ 1198892minus

2119861

(119861 minus 119860)21198872

0) (25)


sides we obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(119861 minus 119860)

2

(1198882+ 1198892)

2 (2119861 minus 119860)le

119861 minus 119860

radic2119861 minus 119860 (26)



119861 minus 119860



we obtain

41198871= (119861 minus 119860) [(119889

2minus 1198882) minus (119889

2

1minus 1198882

1) 119861] = (119861 minus 119860) (119889

2minus 1198882)

(28)


100381610038161003816100381611988711003816100381610038161003816 le

119861 minus 119860

2 (29)


21198871= (119861 minus 119860) (119889

2+ 1198601198892

1minus 2119861119889

2

1) (30)


100381610038161003816100381611988711003816100381610038161003816 le

1

2(119861 minus 119860) (

100381610038161003816100381610038161198892+ 1198601198892

1

10038161003816100381610038161003816+ 2119861

1003816100381610038161003816119889110038161003816100381610038162

)

le1

2(119861 minus 119860) (1 + (119860 minus 1)

1003816100381610038161003816119889110038161003816100381610038162

+ 211986110038161003816100381610038161198891

10038161003816100381610038162

)

=119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

(31)




2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

lt119861 minus 119860

2 (32)


21198871minus 1198872

0= (119861 minus 119860) (

119861

(119861 minus 119860)21198872

0minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (

119861

(119860 minus 119861)21198872

0minus 1198892)

(33)


100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2 (34)


(119899 + 1) 119887119899= (119860 minus 119861) 119888

119899+1 (35)


obtain

10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 (36)







0|2 given by






References



























1 Introductions


119891 (119911) = 119911 +

infin

sum120581=2

119886120581119911120581

(1)


119867[119886 119899] = 119901 isin 119867 119901 (119911) = 119886 + 119886119899119911119899

+119886119899+1

119911119899+1


(2)


119865 (119911) = 119911120573

+

infin

sum120581=2

120582120581119886120581119911120573+120581minus1

120573 isin N

120582120581= 120573120581minus1

119911 =1

120573 120573 |119911| lt 1

(3)


(119891 lowast 119892) (119911) =

infin

sum119896=0

119886119896119887119896119911119896

(4)


119896=0119886119896119911119896 and 119892(119911) = sum

infin

119896=0119887119896119911119896 are analytic in

119880

Let 120595(119911) = 119911120573 + sum

infin



120595 (119911) lowast119911120573

1 minus 120573119911 119911 =

1

120573 120573 isin N 120573 |119911| lt 1 (5)



119876(120485) (120572 120591 120574 120573)

= 119865 (119911) isin 119867 (120573) R119890

times120572[119865(119911)]

(120485)

[119911120573](120485)

+120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

gt 120574 119911 isin 119880

(6)

where

119911 = 0 120573 gt 120485 120485 isin N cup 0 also 120572 + 120591 = 120574 (7)







119891 (119911) =1

120572 + 120591int|119909|


times

infin

sum120581=0

(120572 + 120591) 119909120581

119911120581+1

(120581 + 1) 120591 + 120572] 119889120583 (119909)

(8)




Θ120572120573

(119911) = (120573int119911

0

119905120573minus120572minus1

[119891 (119905)]120572

119892 (119905) 119889119905)

1120573

(10)






119901 (119911) = int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909) (|119911| lt 1) (11)



119892 (119911) +1199111198921015840 (119911)

119888≺ ℎ (119911) (12)

then

119892 (119911) ≺ 119902 (119911) =119888119911minus119888120581

120581int119911

0

119905(119888minus120581)120581

ℎ (119905) 119889119905 ≺ ℎ (119911) (13)



1199111199011015840 (119911)

119901 (119911)≺ ℎ (119911) (14)

then

119901 (119911) ≺ 119902 (119911) = 119886 exp [1

119899int119911

0

ℎ (120585) 120585minus1

119889120585] (15)




(ii) |1199011015840(119911)| le 2R119890119901(119911)(1 minus 1199032)


100381610038161003816100381610038161003816100381610038161003816

1199111199011015840 (119911)

119901 (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 119901 (119911)

(1 minus 119903)2

(16)




(120485)

(120572120591120574120573)




119911120485

[119865 (119911)](120485)

=

120573

119875120485

(120572 + 120591)int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120573 minus 120485) (120572 + 120591) 119909120581119911120573+120581

(120572 + 120591) (120573 minus 120485) + 120581120591] 119889120583 (119909)

(17)

where 119899119875119903




119901 (119911)

=

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

isin 119867 [1 119899]

(18)

ByTheorem 3

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

= int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909)

(19)


[119865 (119911)](120485+1)

[119911120573](120485+1)

+120572[119865 (119911)]

(120485)

120591[119911120573](120485)

=1

120591int|119909|=1

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909119911

1 minus 119909119911119889120583 (119909)

(20)

which yields

119911((120572+120591)(120485minus120573)+120591)120591

int119911

0

[[119865 (120585)]

(120485+1)

[120585120573](120485+1)

+120572[119865 (120585)]

(120485)

120591[120585120573](120485)

]


119889120585

=1

120591int|119909|=1

119911((120572+120591)(120485minus120573)+120591)120591

times int119911

0

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909120585

1 minus 119909120585


119889120585 119889120583 (119909)

(21)



shows that 119865(119911) isin 119876(120485)(120572 120591 120574 120573)



119865 (119911) =1

120572 + 120591int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581119911120573+120581

120572120573 + 120591 (120581 + 120573)] 119889120583 (119909)

(22)


1




10038161003816100381610038161198861205811003816100381610038161003816 le 119867 (120572 120591 120574 120573) (23)

where

119867(120572 120591 120574 120573) =2 (120572 + 120591 minus 120574) (120573 minus 120485) [

120573

119875120485]

120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(24)


119911120485

[119865119909(119911)](120485)

120573119875120485

= 119911120573

+

infin

sum120581=2


(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)(|119909| = 1)

(25)




119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [[120581 minus 1] 119903

120581minus1]

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

le2119903R119890 Ψ (119911)

(1 minus 119903)2

(26)


119866 (119911) equiv120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

+120572[119865(119911)]

(120485)

[119911120573](120485)

minus 120574 (27)

and then119911Ψ1015840 (119911)

Ψ (119911)=

suminfin

120581=2120599 (120572 120591 120574 120573) [120581 minus 1] 119886

120581119911120581minus1

1 + suminfin

120581=2120599 (120572 120591 120574 120573) 119886

120581119911120581minus1

(28)

where

120599 (120572 120591 120574 120573) =120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591 minus 120574) [120573

119875(120485)

]

(29)

From (23) and (28) we got100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [120581 minus 1] 119903

120581minus1

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

(120581 ge 2) (30)


119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 Ψ (119911)

(1 minus 119903)2

(31)

Sinceinfin

sum120581=2


lt 2119903R119890 Ψ (119911)

1 +

infin

sum120581=2

2 (120573 minus 120485) 119903120581minus1

gt (1 minus 119903)2

(32)




(120485)

(120572 120591 120574 120573)



1199111198661015840 (119911)

119866 (119911) ≺ ℎ (119911) (33)

then

119866 (119911) ≺ 119902 (119911) = (120572 + 120591 minus 120574) exp [119899minus1

int119911

0

ℎ (120585) 120585minus1

119889120585 ] (34)



119866 (119911) = (120572 + 120591 minus 120574) +

infin

sum120581=2

120582120581[ [120573+120581minus1]119875

120485+1] 119886120581119911120581minus1

120573119875120485+1

(35)


1199111198661015840

(119911)

119866 (119911) ≺ ℎ (119911) (36)




1199111198661015840 (119911)

119866 (119911) ≺

radic119911

1 minus 119911(37)

then

119866 (119911) ≺ (120572 + 120591 minus 120574) 1 + radic119911

1 minus radic119911

1119899

120572 + 120591 = 120574 (38)




120573 gt 120485 + 1 with 120572 + 120591 = 0 If

119866 (119911) ≺ (120572 + 120591)1 + 119860119911

1 + 119861119911 (39)

then[119865(119911)]

(120485)

[119911120573](120485)

≺(120572 + 120591) (120573 minus 120485)

120591120581119911(120572+120591)(120485minus120573)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(40)


1

(120572 + 120591)119866 (119911) = 1 +

infin

sum120581=2

119862120581119886120581119911120581minus1

(41)

where

119862120581=

120582120581[ (120573+120581minus1)119875

120485+1] sdot [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591) [120573

119875120485+1

] (42)

Let 119892(119911) = [119865(119911)](120485)

[119911120573](120485) then 119892(119911) = 1 + sum

infin

120581=2120603120581119886120581119911120581minus1

where 120603120581= 120582120581[(120573+120581minus1)

119875120485+1

][(120573)

119875120485] and from (39) we have

119892 (119911) +120591

(120572 + 120591) (120573 minus 120485)1199111198921015840

(119911) ≺ (1 + 119860119911

1 + 119861119911) = ℎ (119911) (43)


119892 (119911) ≺(120572 + 120591) (120573 minus 120485)

120591120581119911minus(120572+120591)(120573minus120485)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(44)


Corollary 16 Let

119876(0)

(120572 120591 0 120573) =120572 (119865 (119911))

119911120573+

120591[119865 (119911)]1015840

[119911120573]1015840

(45)

If

119876(0)

(120572 120591 0 120573) ≺ (120572 + 120591) (1 + 119860119911

1 + 119861119911) (46)

then

119865 (119911)

119911120573≺

120572120573 + 120591 (120573 minus 120581)

120591120581119911minus120573(120572+120591)120591120581

times int119911

0

120585(120573(120572+120591)minus120591120581)120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(47)






Acknowledgments



References





















1 Introduction


119891 (119911) =119892 (119911)

ℎ (119911) (1)






R ℎ (119911) gt 0 (119911 isin 119880lowast

) (2)


119891 ≺ 119892 in 119880 or 119891 (119911) ≺ 119892 (119911) (119911 isin 119880lowast

) (3)


|119908 (119911)| le |119911| 119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880lowast

) (4)


119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0)

119891 (119880) sube 119892 (119880) (119911 isin 119880lowast

) (5)





2 Preliminaries


119891 (119911) =1

119911+

infin

sum119899=1

119886119899119911119899

(6)




119891119895(119911) =

1

119911+

infin

sum119899=1

119886119899119895119911119899

(7)


and 1198912(119911) by

(1198911lowast 1198912) =

1

119911+

infin

sum119899=1

11988611989911198861198992119911119899

(8)


119902120582120583

(119911) =1

119911+

infin

sum119899=1

(120582

119899 + 1 + 120582)

120583

119911119899

(120582 gt 0 120583 ge 0) (9)



119871120582120583119891 (119911) = (119891 (119911) lowast 119902

120582120583(119911))

=1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(10)


119905119886119892 and 119871


tively by

119871120582120583119892 (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198871198991003816100381610038161003816 119911119899


119871120582120583ℎ (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198881198991003816100381610038161003816 119911119899


ℎ (119911) isin Σ)

(11)


119871120582120583

(119891 lowast 119892) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast)

119871120582120583

(119891 lowast ℎ) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198881198991003816100381610038161003816 119911119899


)

(12)






property

120588119871120582120583

(119891 lowast 119892) (119911)

119871120582120583

(119891 lowast ℎ) (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911 (119911 isin 119880

lowast

) (13)


and 119871(120582 120583)(119891 lowast ℎ)(119911) = 0


120601 (120572 120573 119911) =1

119911+

infin

sum119899=0

(120572)119899+1

(120573)119899+1

119911119899

(14)



120601 (120572 120573 119911) =1

119911 21198651(1 120572 120573 119911) (15)

where

21198651(119887 120572 120573 119911) =

infin

sum119899=0

(119887)119899(120572)119899

(120573)119899

119911119899

119899(16)


120582120583(119911)


119871 (120572 120573 120582 120583) 119891 (119911) = (119891 (119911) lowast 120601 (120572 120573 119911) lowast 119902120582120583

(119911))

=1

119911+

infin

sum119899=1

(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(17)




119911(119871 (120572 120573 120582 120583) 119891 (119911))1015840

= 120572119871 (120572 + 1 120573 120582 120583) 119891 (119911)

minus (120572 + 1) 119871 (120572 120573 120582 120583) 119891 (119911) (18)


120588119871 (120572 + 1 120573 120582 120583) 119891 (119911)

119871 (120572 120573 120582 120583) 119891 (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911

(0 le 119861 lt 119860 le 1 120588 gt 0)

(19)



relation (19) above






infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861)

minus10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(20)


119891119899(119911) =

1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

(21)



1003816100381610038161003816100381610038161003816100381610038161003816

120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899) 119911119899+1

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899119911119899+1

)

minus11003816100381610038161003816100381610038161003816100381610038161003816

le 120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899)10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899

10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816)

minus1

le 1

(22)





100381610038161003816100381610038161003816100381610038161003816lt 1

(119911 isin 119880lowast)

(23)


Σ120583




120582(120588 119860 119861) then

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119899 ge 1

(24)


(21)


120582(120588 119860 119861)



1

119903minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903

le1003816100381610038161003816119891 (119911)

1003816100381610038161003816

le1

119903+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903


1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

le100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(25)



inequalityinfin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(26)


1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

le1

119903+ 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

119903+ 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|minus

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

ge1

119903minus 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

ge1

119903minus 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(27)


sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(119860 minus 119861) (2 + 120582)

120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(28)

Hence

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2+

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

le1

1199032+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

ge1

1199032minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816

ge1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(29)



120582(120588 119860 119861) which




disk |119911| lt 1199031 where

1199031= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(30)



119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (31)

For |119911| lt 1199031 we have

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 1 + 120582))

120583

119886119899119911119899

1119911 + suminfin

119899=1(120582(119899 + 1 + 120582))

120583

119886119899119911119899

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

100381610038161003816100381610038161003816100381610038161003816

lesuminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(32)


sum119899=1

(119899 + 1) (120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1


infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

)

(33)

or

suminfin

119899=1(119899 + 2 minus 120575) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(1 minus 120575)le 1 (34)


(119899 + 2 minus 120575) (120582(119899 + 1 + 120582))120583

|119911|119899+1

(1 minus 120575)

le (120582

119899 + 120582 + 1)

120583

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899 ge 1

(35)



|119911| le (1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899+1

(36)




disk |119911| lt 1199032 where

1199032= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(37)



1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891 (119911))10158401015840

(119891 (119911))1015840+ 2

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (38)






infin

sum119899=1

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(39)


119891119899(119911)

=1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

119899 ge 1

(40)




120582(120588 120572 120573 119860 119861) then1003816100381610038161003816119886119899

1003816100381610038161003816

le(119860 minus 119861) (119899 + 120582 + 1)

120583

(120573)119899+1

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861)) (120572)

119899+1

119899 ge 1

(41)


(40)




1199033= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(42)





1199034= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(43)






Acknowledgment


References






















1 Introduction


Φ (119911 119904 119886) =

infin

sum119899=0

119911119899

(119899 + 119886)119904


0 119904 isin C when |119911| lt 1

R (119904) gt 1 when |119911| = 1)

(1)


119904(120585) defined by

120577 (119904) =

infin

sum119899=1

1

119899119904= Φ (1 119904 1) = 120577 (119904 1) (R (119904) gt 1)

120577 (119904 119886) =

infin

sum119899=0

1

(119899 + 119886)119904


0)

ℓ119904(120585) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 1)119904= Φ (e2120587119894120585 119904 1) (R (119904) gt 1 120585 isin R)

(2)


119904(119911)

Li119904(119911) =

infin

sum119899=1

119911119899

119899119904= 119911Φ (119911 119904 1)


(3)


120601 (120585 119886 119904) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 119886)119904= Φ (e2120587119894120585 119904 119886)



when 120585 isin Z)

(4)




Φ (119911 119904 119886) =1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

1 minus 119911eminus119905d119905

(R (119886) gt 0R (119904) gt 0 when |119911| ≦ 1 (119911 = 1) R (119904) gt 1

when 119911 = 1) (5)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) =

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

119911119899

(119899 + 119886)119904

(119901 119902 isin N0 120582

119895isin C (119895 = 1 119901)

119886 120583119895isin C Z

minus

0(119895 = 1 119902)

120588119895 120590

119896isin R

+

(119895 = 1 119901 119896 = 1 119902)





lowast

)

(6)

with

nablalowast

= (

119901

prod119895=1

120588minus120588119895

119895) sdot (

119902

prod119895=1

120590120590119895

119895)

Δ =

119902

sum119895=1

120590119895minus

119901

sum119895=1

120588119895

Ξ = 119904 +

119902

sum119895=1

120583119895minus

119901

sum119895=1

120582119895+119901 minus 119902

2

(7)



(120582)120581=Γ (120582 + 120581)

Γ (120582)

=


1 (120581 = 0 120582 isin C 0)

(8)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0)

(9)



119898or

119897Ψ119898(119897 119898 isin N

0) involved


119897119865119898(119897 119898 isin N




1198871 119887

119898such that

119886119895isin C (119895 = 1 119897) 119887

119895isin C Z

minus

0(119895 = 1 119898)

(10)


119897Ψlowast

119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

=

infin

sum119899=0

(1198861)1198601119899sdot sdot sdot (119886

119897)119860119897119899

(1198871)1198611119899sdot sdot sdot (119887

119898)119861119898119899

119911119899

119899

=Γ (119887


119898)

Γ (1198861) sdot sdot sdot Γ (119886

119897) 119897Ψ119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

(119860119895gt 0 (119895 = 1 119897) 119861

119895gt 0 (119895 = 1 119898)

1 +

119898

sum119895=1

119861119895minus

119897

sum119895=1

119860119895≧ 0)

(11)


|119911| lt nabla = (

119897

prod119895=1

119860minus119860119895

119895) sdot (

119898

prod119895=1

119861119861119895

119895) (12)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

Γ (119904)intinfin

0


119905120582)

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0)

(13)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 0 120582)

= Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

= e119887Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 0)

(14)


119901 minus 1 = 119902 = 0 (1205821= 120583 120588

1= 1) (15)


Θ120582

120583(119911 119904 119886 119887)

=1

Γ (119904)intinfin

0


119905120582) (1 minus 119911eminus119905)

minus120583

d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0 120583 isin C)

(16)


R (119904) gt 0 when 119887 = 0 |119911| ≦ 1 (119911 = 1) (17)

or

R (119904 minus 120583) gt 0 when 119887 = 0 119911 = 1 (18)




Θ120582


Θ120582

120583(119911 119904 119886 0) = Φ

lowast

120583(119911 119904 119886) =

1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

(1 minus 119911eminus119905)120583d119905

(R (119886) gt 0 R (119904) gt 0 when |119911| ≦ 1 (119911 = 1)

R (119904 minus 120583) gt 0 when 119911 = 1)

(19)



[4] as follows

Φlowast

120583(119911 119904 119886) =

infin

sum119899=0

(120583)119899

(119886 + 119899)119904

119911119899

119899 (20)


120583(119911 119904 119886 119887) that is


Θ1

1(1 119904 119886 119887)

= 120577119887(119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1


119905) d119905

(21)




(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899

119899(120582 gt 0)

(22)



119867119898119899

pq (119911)

= 119867119898119899

pq [119911 |(119886

1 119860

1) (119886p 119860p)

(1198871 119861

1) (119887q 119861q)

]

=1

2120587119894intL

Ξ (119904) 119911minus119904d119904 (119911 isin C 0

1003816100381610038161003816arg (119911)1003816100381610038161003816 lt 120587)

(23)

where

Ξ (119904) =prod

119898

119895=1Γ (119887

119895+ 119861

119895119904) sdot prod

119899

119895=1Γ (1 minus 119886

119895minus 119860

119895119904)

prodp

119895=119899+1Γ (119886

119895+ 119860

119895119904) sdot prod

q

119895=119898+1Γ (1 minus 119887

119895minus 119861

119895119904)

(24)


119895gt 0 (119895 =

1 p) 119861119895gt 0 (119895 = 1 q) 119886

119895isin C (119895 = 1 p)



Γ (119887119895+ 119861

119895119904)

119898

119895=1

(25)


Γ (1 minus 119886119895+ 119860

119895119904)

119899

119895=1

(26)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902





1 minus 119905)

=

infin

sum119899=0

119871(120572)

119899(119909) 119905

119899

(|119905| lt 1 120572 isin C)

(27)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=eminus119887

Γ (119904)

infin

sum119899=0

119899

sum119896=0

(minus1)119896

(119899

119896) sdot Γ (119904 + 120582 (120572 + 119896 + 1))

sdot 119871(120572)

119899(119887)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 + 120582 (120572 + 119895 + 1) 119886)

(R (119886) gt 0 R (119904 + 120582120572) gt minus120582)

(28)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) (29)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



D120572

119911119911

119901

119891 (119911) =1

Γ (minus120572)int119911

0

119891 (120585) 120585119901


(R (120572) lt 0 R (119901) gt 1)

(30)


D120572

119911119911

119901

119891 (119911) =d119898

d119911119898D

120572minus119898

119911119911

119901

119891 (119911) (31)



Im(120585)

Branch line for

Branch line for


exp[p(ln(g(120585)))]

C1

C2

C3

C4

Re(120585)

z

a

gminus1(0)





119901


119863120572

119892(119911)[119892 (119911)]

119901

119891 (119911)

=eminus119894120587119901Γ (1 + 120572)4120587 sin (120587119901)

times int119862(119911+119892

minus1(0)+119911minus119892

minus1(0)minus119865(119886)119865(119886))

sdot119891 (120585) [119892 (120585)]

119901

1198921015840 (120585)

[119892 (120585) minus 119892 (119911)]120572+1

d120585

(32)



1cup119862

2cup119862

3cup119862








119892(119911)[119892(119911)]

119901



119863120572

119911119911

119901

=Γ (1 + 119901)

Γ (1 + 119901 minus 120572)119911119901minus120572

(R (119901) gt minus1) (33)



11205881199011205901120590119902)

12058211205821199011205831120583119902


119863120572

119911119911

120573minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899)

1198871120582

| (119904 1) (01

120582)]119863

120572

119911119911

120573minus1119911119899

119899

=Γ (120573)

120582Γ (119904) Γ (120573 minus 120572)

infin

sum119899=0

(120573)119899

(120573 minus 120572)119899

sdot

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899+120573minus120572minus1

119899

=Γ (120573)


Φ(1205881120588119901112059011205901199021)

12058211205821199011205731205831120583119902120573minus120572

(119911 119904 119886 119887 120582)







119863120572

119892(119911)119891 (119911) = 119863

120572

ℎ(119911)119891 (119911) 119892

1015840(119911)

ℎ1015840(119911)(ℎ(119908) minus ℎ(119911)

119892(119908) minus 119892(119911))

120572+1

100381610038161003816100381610038161003816100381610038161003816119908=119911

(35)




119911119901119874120572

120573119891 (119911)

= 119901(119911119901minus1

)minus120572

sdot119911

119874120572

120573119891 (119911) (119911

119901minus1

)120572

119901minus1

prod119904=1

(1 minus119911

119908eminus(2120587119894119904)119901)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(36)

where

119892(119911)119874120572


Γ (120573)

Γ (120572)[119892 (119911)]

1minus120573

119863120572minus120573

119892(119911)[119892 (119911)]

120572minus1

sdot sdot sdot

(37)



120579 (119911) = (119911 minus 1199110) 119902 (119911) (38)


119870

=0 1 [119888] ([119888] 119905ℎ119890 119897119886119903119892119890119904119905 119894119899119905119890119892119890119903 119899119900119905 119892119903119890119886119905119890119903 119905ℎ119886119899 119888)

(39)


0and let

120596 = exp (2120587119894119886) (40)



sum119896isin119870

119888minus1

120596minus120574119896

119891 (120579minus1

(120579 (119911) 120596119896

))

=

infin


[120579 (119911)]119888119899+120574

Γ (119888119899 + 120574 + 1)

sdot119863119888119899+120574

119911minus119887119891 (119911) 120579

1015840

(119911) (119911minus119911

0

120579 (119911))

119888119899+120574+1

100381610038161003816100381610038161003816100381610038161003816119911=1199110

(1003816100381610038161003816119911minus1199110

1003816100381610038161003816=100381610038161003816100381611991101003816100381610038161003816)

(41)



119891 (119911) = 119888

infin


(119911 minus 1199110)119888119899+120574

Γ(119888119899 + 120574 + 1)119863

119888119899+120574

119911minus119887119891(119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199110

(42)



1)(119911minus119911

2)) where 119911

1and 119911

2are two



120596 = exp(2120587119894119886) (43)


1and 119911


curves

119862 (119905) 119862 (119905) subR 0 lt 119905 ≦ 119903 (44)

be defined by

119862 (119905) = 1198621(119905) cup 119862

2(119905)

= 119911 1003816100381610038161003816120582119905 (1199111 1199112 119911)

1003816100381610038161003816 =1003816100381610038161003816100381610038161003816120582119905(119911

1 119911

21199111+ 119911

2

2)1003816100381610038161003816100381610038161003816

(45)

where

120582119905(119911

1 119911

2 119911) = [119911 minus

1199111+ 119911

2

2+ 119905 (

1199111minus 119911

2

2)]

sdot [119911 minus (1199111+ 119911

2

2) minus 119905 (

1199111minus 119911

2

2)]

(46)


1+ 119911



1199111+ 119911

2

2+ (

1199111minus 119911

2

2) 119905 (47)


2

2minus (

1199111minus 119911

2

2) 119905 (48)

Im(120585)

Branch line for

Branch line for

C1

C2

z1

z2

Re(120585)

z

c =z1 + z2

2

(120585 minus z2)120583120579(120585)

aminus120574minus1

(120585 minus z1)120579(120585)

aminus120574minus1



[(119911 minus 1199111) (119911 minus 119911

2)]

120582


2))))

(49)


plusmn

defined by

119871plusmn=

119911 119911 =1199111+ 119911

2

2plusmn 119905 (

1199111minus 119911

2

2) (0 ≦ 119905 ≦ 1)

119911 119911 =1199111+ 119911

2

2plusmn 119894119905 (

1199111minus 119911

2

2) (119905 lt 0)

(50)



1)(119911minus119911

2) gt 0 (v)


2)119901


+cup 119871


119911minus1199112

(119911 minus 1199112)119901


119870 = 119896 119896 isin N arg(120582119905(119911

1 119911

21199111+ 119911

2

2))

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) +

2120587119896

119886

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) + 2120587

(51)


defined by

120585 =1199111+ 119911

2

2+1199111minus 119911

2


sum119896isin119870

119888minus1120596minus120574119896

1199111minus 119911

2

119891 (120601minus1

(120596119896

120601 (119911)))

times [120601minus1

(120596119896

120601(119911)) minus 1199111]][120601

minus1

(120596119896

120601(119911)) minus 1199112]120583

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

[120601 (119911)]119888119899+120574

(52)


where

120601 (119911) =119911 minus 119911

1

119911 minus 1199112

(53)



2)120583

1199111minus 119911

2

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

(119911 minus 119911

1

119911 minus 1199112

)

119888119899+120574

(54)


2) where 119911

1and 119911




sum119896isin119870

119888minus1

120596minus120574119896

[(1199112minus 119911

1+ radicΔ

119896

2)

120572

(1199111minus 119911

2+ radicΔ

119896

2)

120573

sdot 119891 (1199111+ 119911

2+ radicΔ

119896

2)


sdot (1199112minus 119911

1minus radicΔ

119896

2)

120572


2minus radicΔ

119896

2)

120573

119891(1199111+ 119911

2minus radicΔ

119896

2)]

=

infin



eminus119894120587119888(119899+1)[120579 (119911)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911minus1199112


(120579 (119911)

(119911minus1199112) (119911minus119911

1))


1205791015840

(119911)119891 (119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199111

(55)

where

Δ119896= (119911

1minus 119911

2)2

+ 4119881 (120596119896

120579 (119911))

119881 (119911) =

infin

sum119903=1

119863119903minus1

119911[119902 (119911)]

minus119903

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

119911119903

119903

120579 (119911) = (119911 minus 1199111) (119911 minus 119911

2) 119902 ((119911 minus 119911

1) (119911 minus 119911

2))

(56)


(119911 minus 1199111)(119911 minus 119911

2)) and 119911


119891 (119911) = 119888119911minus120573

(119911 minus 1199111)minus120572

times

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot119863minus120572+119888119899+120574

119911119911


(119911 + 119908 minus 1199111)119891(119911)

10038161003816100381610038161003816119911=1199111

(119908=119911)

(57)



119863120572

119911119911

119901+119902

119906 (119911) V (119911)

=

infin


(120572

120574 + 119899)119863

120572minus120574minus119899

119911119911

119901

119906 (119911)119863120574+119899

119911119911

119902V (119911) (58)




119863120572

119911119911

120572+120573minus1

119906 (119911) V (119911)


sdot

infin


119863120572+]+1minus119899119911



(60)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 (1120582))]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot 119865(119896minus1)



where 120582 gt 0 and 119865(119899)119863


119865(119899)

119863[119886 119887

1 119887

119899 119888 119909

1 119909

119899]

=

infin

sum1198981119898119899=0

(119886)1198981+sdotsdotsdot+119898

119899

(1198871)1198981

sdot sdot sdot (119887119899)119898119899

(119888)1198981+sdotsdotsdot+119898

119899

1199091198981

1


119909119898119899

119899

119898119899

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091198991003816100381610038161003816 lt 1)

(62)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119896

(119911119896minus1)120572

sdot119911119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) (119911119896minus1

)120572

sdot

119896minus1

prod119904=1

(1 minus119911

119908eminus2120587119894119904119896)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(63)




119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

(64)



119896

(119911119896minus1)120572

119911

119874120572

120573

times

1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119911119899+(119896minus1)120572

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


sdot((119911119908) eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119908=119911

=119896

120582Γ (119904) (119911119896minus1)120572

times

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981



sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

times119911

119874120572

120573119911


119896minus1+119899+(119896minus1)120572

=119896

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


times(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

sdotΓ (120573) Γ (119898


119896minus1+ 119899 + 119896120572)

Γ (120572) Γ (120573 + 1198981+ sdot sdot sdot + 119898

119896minus1+ 119899 + 119896120572)


=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 1120582)]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(119899+119896120572)1198981+sdotsdotsdot+119898

119896minus1

(1+120572minus120573)1198981



119896minus1

sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

(65)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


119911minus1198881198990(119911 minus 119911

0)119888119899

Γ (119888119899 + 1) Γ (1 minus 119888119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(1003816100381610038161003816119911 minus 1199110

1003816100381610038161003816 =100381610038161003816100381611991101003816100381610038161003816 120582 gt 0)

(66)


Proof Setting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


0 we

have

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


(119911 minus 1199110)119888119899

Γ (1 + 119888119899)

sdot119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

(67)


0| = |119911

0|


119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

=119911minus1198881198990

Γ (1 minus 119888119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(68)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(69)


119911 =1199111

2+1199111



Proof By taking 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 9 with 119911

2= 0 120583 = 120572 ] = 120573 and 0 lt 119888 ≦ 1 we find

that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888(119911 minus 1199111)minus120573

119911minus120572

1199111

sdot

infin



sdot (119911 minus 119911

1

119911)119888119899+120574

sdot119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

(71)



119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120572+120573minus1

1

Γ (120572 + 119888119899 + 120574)

Γ (120572 + 120573)

times Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582)

(72)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573+120574

(119911 minus 1199111)minus120572+120574

119911120573+120572minus2120574minus1

1

sdot

infin


sin [(120573 minus 119888119899 minus 120574) 120587] e119894120587119888(119899+1)

sin [(120573 + 119888 minus 120574) 120587] Γ (1 minus 120572 + 119888119899 + 120574)

sdot (119911(119911 minus 119911

1)

11991121

)

119888119899

Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdot [(119911 minus 1199111)Φ

(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (120573 minus 119888119899 minus 120574

120572 + 120573 minus 2119888119899 minus 2120574) 119911

1

sdotΦ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)]

(73)


119911 =1199111

2+1199111



Proof Putting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 10 with 119911

2= 0 0 lt 119888 ≦ 1 119902(119911) = 1 and 120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2) we find that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573

(119911 minus 1199111)minus120572

sdot

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

(75)


119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

= 119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

+ (119911 minus 1199111)119863

minus120572+119888119899+120574

119911


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120573+120572minus2119888119899minus2120574

1


Γ (1 + 120573 + 120572 minus 2119888119899 minus 2120574)

sdot Φ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (119911 minus 119911

1

1199111

)Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdotΦ(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582) )

(76)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

=Γ (120591) Γ (1 + ] minus 120591) sin (120574120587)

120587

sdot

infin


(minus1)119899

Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)


(77)



12058211205821199011205831120583119902


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=

infin


(] minus 120591120574 + 119899


119911]minus1

sdot 119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(78)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)



119911]minus1 =

Γ (])Γ (120591 + 120574 + 119899)

119911120591+120574+119899minus1

119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911minus120574minus119899

Γ (1 minus 120574 minus 119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(79)



Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

= (Γ (120591) Γ (1 + ] minus 120591) sin (120573120587)




sdotsin (120579120587)

sin [(120579 + 120574) 120587]

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(80)



120572 = ] minus 120591 119906 (119911) = 119911120591minus120573

V (119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(81)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


sdot

infin


119863]minus120591+120574+1minus119899119911




119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(82)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

119863]minus120591+120574+1minus119899119911


=Γ (] minus 120579 minus 119899)


119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (120579 + 119899)

Γ (1 + 120579 + 120574)

times 119911120579+120574

Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(83)





lim119887rarr0

11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)] = 120582Γ (119904)

(120582 gt 0)

(84)



Φ(120588112058811990113120590

112059011990213)

12058211205821199011205721205831120583119902120573

(119911 119904 119886)

=3Γ (120573) Γ (3120572)

Γ (120572) Γ (120573 + 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

(3120572)119899

(120573 + 120572)119899

119911119899

(119899 + 119886)119904


(85)


1198651[119886 119887

1 119887

2 119888 119909

1 119909

2]

=

infin

sum11989811198982=0

(119886)1198981+1198982

(1198871)1198981

(1198872)1198982

(119888)1198981+1198982

1199091198981

1

1198981

1199091198982

2

1198982

(max 100381610038161003816100381611990911003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 lt 1)

(86)




1= 120583 in



Θ120582

120583(119911 119904 119886 119887)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(111)

120583120572+119888119899+120574120572+120573(119911

1 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(87)


119911 =1199111

2+1199111





Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886)

=Γ (120591) Γ (1 + ] minus 120591)



times (Γ (1 + 120574 + 120579) sin [(] minus 120591 + 120573) 120587]


sdot

infin


Γ (]minus120579minus119899) Γ (120579+119899)Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120582

(119911 119904 119886)





References















































Algebra




Geometry












Operations Research








Contents





















Toka Diagana










1 Introduction




119886(119899 119896) and 119878



(119905)119899=

119899

sum119896=0

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

(119905 minus 119886)119896

(1)

(119905 minus 119886)119899

=

119899

sum119896=0

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(119905)119896 (2)


119904119886(119899 119896) =

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

119878119886(119899 119896) =

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(3)


119904119886(119899 + 1 119896) = 119904

119886(119899 119896 minus 1) + (119886 minus 119899) 119904

119886(119899 119896) (4)

119878119886(119899 + 1 119896) = 119878

119886(119899 119896 minus 1) + (119896 minus 119886) 119878

119886(119899 119896) (5)


119904119886(0 0) = 1 119904

119886(119899 0) = (119886)

119899 119904

119886(0 119896) = 0 119899 119896 = 0

119878119886(0 0) = 1 119878

119886(119899 0) = (minus119886)

119899

119878119886(0 119896) = 0 119899 119896 = 0

(6)


1+ 119883


119896of





119875 [119882 = 119899] =119896

(1 minus 120579)minus119904

(minus log (1 minus 120579))119896

120579119899

119899(minus1)

119899minus119896

119904minus119904(119899 119896)

119875 [119885 = 119899] =119896

119890119898 (119890119897 minus 1)119896

119897119899

119899(minus1)

119899minus119896

119878minus119898119897

(119899 119896)

(7)






119870(1)

119899119903= 119903 + [log(119899 minus 1


1

119903+ 119900 (1)] (8)


2 Explicit Formula



119865119899(119909) = prod

119899


1198651015840

119899(119909) = sum


119899minus1le119899

119894isinN11989911989511198952119895119899minus1

1198911015840

119894(119909)

119899minus1

prod119896=1

119891119895119896(119909) (9)


1 2

119889

119889120585(120585119895 + 119886)

1= 119895

119889

119889120585(120585119895 + 119886)

2= (120585119895 + 119886) 119895 + (120585119895 + 119886 minus 1) 119895

(10)


1

119895119899119889

119889120585(120585119895 + 119886)



119899minus1

prod119896=1

(120585 +119886 minus 119895

119896

119895) (11)



1

119895119899119889119896

119889120585119896(120585119895 + 119886)


119899minus119896le119899minus1

119896

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(12)


1

119895119899119889119898

119889120585119898(120585119895 + 119886)


119899minus119898le119899minus1

119898

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(13)

Then

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899


119899minus119898le119899minus1

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(14)




119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895) = sum



prod119902=1

(120585 +119886 minus 119894

119902

119895)

(15)

119894119902

isin 1198951 1198952 119895

119899minus119898 has ( 119899minus119898





119902=1(120585 + (119886 minus





119895119899minus119898minus1





119902=1(120585 + (119886 minus 119895


of (1119895119899)(119889119898+1119889120585119898+1)(120585119895 + 119886)119899 Thus we have

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899



119898 + 1

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)



119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(16)

Lemma 2 Let 119904(119899 119896 119886) =

(1119896)lim120585rarr0

((sum119896

119895=0(minus1)

119896minus119895

( 119896119895) (119889119896119889120585119896)(120585119895 + 119886)

119899)119896)

Then


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (17)

Proof Using Lemma 1

119904 (119899 119896 119886)

=1

119896lim120585rarr0

((

119896

sum119895=0


(119896

119895) 119895119899


119899minus119896le119899minus1

119896

times

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)) (119896)

minus1

)

(18)



(120585 + (119886 minus 119895119902)119895) = (1119895119899minus119896)prod

119899minus119896

119902=1(120585119895 + 119886 minus 119895

119902)


1

119896lim120585rarr0

[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(120585119895 + 119886 minus 119895119902)]

]

=1

119896[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)]

]

(19)

which boils down to


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (20)

since

1

119896

119896

sum119895=0


(119896

119895) 119895119896

= 119878 (119896 119896) = 1 (21)



119904119886(119899 119896) = 119904 (119899 119896 119886) = sum


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (22)

Proof We know that


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (23)



119899+1minus119896)







119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (24)

is equal to


119899minus119896+1le119899minus1

119899minus119896+1

prod119902=1

(119886 minus 119895119902) + (119886 minus 119899)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)

(25)

This implies that

119904 (119899 + 1 119896 119886) = 119904 (119899 119896 minus 1 119886) + (119886 minus 119899) 119904 (119899 119896 119886) (26)



119886(119899 119896)


120582 = (1205821ge 120582


119896) (27)


119894


14= 11989115

= 11989123

= 11989131

= 11989142

= 1 119891119894119895= 0 elsewhere

(1 le 119895 le 120582119894)

0 0 0 1 10 0 11 0 00 10

(28)



(119886119894)119894ge0



1198860 1198861 119886

ℎ then |119879119889119860(ℎ 119903)| = ( ℎ+1




120596-tableau is











119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)


120596 (|119888|) (29)




is equal to 119904119886(119899 119896)



0 1 2 119899 minus 1 and

120596119860(120601) = prod

119888isin120601

120596 (|119888|)

=

119899minus119896

prod119894=1

(119886 minus 119895119894) 119895

119894isin 0 1 2 119899 minus 1

(30)

If 119886 = 1198861+ 1198862for some 119886

1and 119886

2 then with 120596lowast(119895) = 119886

2minus 119895

120596119860(120601) =

119899minus119896

prod119894=1

(1198861+ 120596lowast

(119895119894))

=

119899minus119896

sum119903=0

119886119899minus119896minus119903

1sum


119903le119895119899minus119896

119903

prod119894=1

120596lowast

(119902119894)

(31)






may write

120596119860(120601) = sum

120595isin119861120601

120596119860(120595) (32)


120596119860(120595) = 119886

119899minus119896minus119903

1

119903

prod119894=1

120596lowast

(119902119894) (33)

where 1199021 1199022 119902

119903isin 119895

1 1198952 119895


written as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

sum120595isin119861120601

120596119860(120595) (34)



1 1198952 119895

119899minus119896



(119899

119896)(

119899 minus 119896

119903) (35)





( 119899119896) ( 119899minus119896

119903)

( 119899119903)

= (119899 minus 119903

119896) (36)


119904119886(119899 119896) =

119899minus119896

sum119903=0

(119899 minus 119903

119896) 119886119899minus119896minus119903

1sum120595isin119861119903


120596lowast

(|119888|) (37)



119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

1sum



120596lowast

(|119888|) (38)

Clearly 119861119899minus119895



tity

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

11199041198862

(119899 119895) (39)


1and 119886

2





119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (40)






2isin 1198791198891198602(119895 minus 1 119895 minus 119899+ 119896)


2 we obtain an


119860

(119898 +


sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601)

=

119899

sum119896=0

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011)

times

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

(41)


Note that

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)


119895minus119899+119896le119898+119895minus1

119895minus119899+119896

prod119902=1

(119886 minus 119892119902)



119895minus119899+119896

prod119902=1

(119886 minus (119898 + 119892119902))

= 119904119886minus119898

(119895 119899 minus 119896)

(42)


sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011) = 119904

119886(119898 119896)

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601) = 119904

119886(119898 + 119895 119899)

(43)

Thus

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (44)



convolution formula

119904119886(119899 + 1119898 + 119895 + 1) =

119899

sum119896=0

119904119886(119896 119899) 119904

119886minus(119896+1)(119899 minus 119896 119895) (45)

Proof Let


in 1198601= 0 1 119896 minus 1


are in 1198602= 119896 + 1 119899








21199092 + sdot sdot sdot + 119886

119899119909119899 has only

real zeros then

1198862

119896ge 119886119896+1

119886119896minus1

119896

119896 minus 1

119899 minus 119896 + 1

119899 minus 119896(119896 = 2 119899 minus 1) (46)


sum119896=0

119904119886(119899 119896) (119905 + 119886)

119896

(47)



119896=0119904119886(119899 119896)119905119896 are 119886 119886minus1 119886minus



119899


and hence unimodal


⟨119905⟩119899=

119899

sum119896=0


119904119886(119899 119896) (119905 + 119886)

119896

(48)



119904119886(119899 119896) = (minus1)

119899minus119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902) (49)


119886(119899 119896)| as

1003816100381610038161003816119904119886 (119899 119896)1003816100381610038161003816 = (minus1)

119899minus119896

119904119886(119899 119896)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902)

(50)



lt 1199062


infin

sum119894=1

1

119906119894

= infin

infin

sum119894=1

1

1199062119894

lt infin (51)




which sum119899119896


119870119899= 119899 minus [

119899

sum119894=1

1

119906119894

minus

119899

sum119894=1

1

1199062119894

(1 +1

119906119894

)

minus1

+ 119900 (1)] (52)


1

1+1

2+ sdot sdot sdot +

1

119899= log 119899 + 120574 + 119900 (1) (53)



119886(119899 119896)|

119899

119896=0



119899


mation

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (54)




Denoting by 119896119886119899



1= 119887 + 0 119906

2= 119887 + 1 119906


119896119886119899

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894)2(1 +

1

119887 + 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894) (119887 + 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894 + 1+ 119900 (1)]

(55)


119899minus1

sum119894=0

1

119887 + 119894 + 1= log (119887 + 119899) minus log 119887 (56)

From this we get

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (57)



119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (58)



1)(119886 minus


119899minus119896)where 119895


1 By Theorem 10 119904119886(119899 119896) = sum

119899119899minus119896 Thus with 119906

1= 119886 119906

2=

119886 minus 1 119906119899minus1

= 119886 minus 119899 + 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894)2(1 +

1

119886 minus 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894) (119886 minus 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119886 minus 119894 + 1+ 119900 (1)]

(59)


119899119896 0 1 2 30 11 12 2 13 6 6 14 24 35 10 15 120 225 85 156 720 1624 735 1757 5040 13132 6769 19608 40320 109584 118124 672849 362880 1026576 1172700 72368010 3628800 10628640 12753576 8409500


119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (60)


9


119896minus19

= [log(1 + 8

1) + 119900 (1)]

= [log 9 + 119900 (1)]

asymp 2

(61)


10


1198961010

= [log( 10 + 1

10 minus 10 + 1) + 119900 (1)]

= [log 11 + 119900 (1)]

asymp 2

(62)







(63)


Table 2 Values of 11990410(119899 119896)

119899119896 0 1 2 30 11 10 12 90 19 13 720 242 27 14 5040 2414 431 345 30240 19524 5000 6356 151200 127860 44524 81757 604800 662640 305956 772248 1814400 2592720 1580508 5376289 3628800 6999840 5753736 265576410 3628800 10628640 12753576 8409500


119904(119899 119896) 0 1 2 3 4 50 11 0 12 0 1 13 0 2 3 14 0 6 11 6 15 0 24 50 35 10 16 0 120 274 225 85 157 0 720 1764 1624 735 1758 0 540 13068 13132 6769 19609 0 40320 109584 118124 67284 2244910 0 362880 1026576 1172700 723680 269325





2= 2 119906


119896119899= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

1

1198942(1 +

1

119894)minus1

+ 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

(1

119894minus

1

119894 + 1) + 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894 + 1+ 119900 (1)]

(64)


119899minus1

sum119894=1

1

1 + 119894= log 119899 minus log 1 + 120574 + 119900 (1) (65)

Therefore we have

119896119899= [log 119899 + 120574 + 119900 (1)] (66)



7

119896=0occurs at

1198967= [log 7 + 120574 + 119900 (1)]

= [195 + 05772 + 119900 (1)]

= [253 + 119900 (1)]

asymp 2

(67)


11989610

= [log 10 + 120574 + 119900 (1)]

= [230 + 05772 + 119900 (1)]

= [28772 + 119900 (1)]

asymp 3

(68)




Acknowledgment


References
















1 Introduction















2 New Instructions







x

f(x)

i

(ii) and (iii)1

c













2119888 and



















2minus119890


minus119873

2minus119873

] (1)









2119909 multiplica-








1sdot finite

1+ finite

2= 2 (2)








2120583 in the case






2((maximum



Inputsign exp mant

Drop leading zeros

numZeros + 1


Add

Switch

s e12 m51

lookup output


exp

Add 1

expP1

adjustedE

Switch

bitRot


Add 0x7ff

Add 0x3ff

Switch

Concatenate bit

xor

Add drop carry

moreThan22Zeros

isNot12

Concatenate bit

51

42

42 + 1

44

30

31

52

52

52

52


Add



(c)

(b)

(d)

(e)

(f)

(g)


(h)

(i)

(j)

(k)

leading12Mant

(a)

(p)

(r)

(o)

(l)

(m)

(n)

(q)

(s)

(t)

(u)

(v)

5

sim

sim

0x7ffSubs

Subs 0b0 0




12-bit lookup

12-bit lookup

ge1

e(last 10bits)lt1

sim


0b10 0Subs

Subs 0b0 0

low 10bits4

6

11

11

11

11

12

Lower 10






2119872

recip 3 113 376 0500 2048 3 infin






3 Evaluation





4 Conclusion




(a)


recip (2minus119890

119888)ext (2minus119890



119888)sat

sqrt (2minus119890

119888)ext 21198902

119888 0infin 0 NaN 0 0 0 NaN




(b)


minusinfin 119888 119888 0 0

infin 119888 0 119888 0

NaN 119888 119888 119888 0

(c)



minusinfin119891

minusinfin119891

infin infin119891

infin119891

infin119891

infin119891

NaN0

NaN119891

NaN119891

NaN119891

NaN119891

NaN1

2119891

2119891

2119891

2119891

NaN2

1radic2119891

1radic2119891

1radic2119891

1radic2119891

NaN3

0119891

0119891

0119891

0119891

recipdiv

sqrtrsqrtcbrt

rcbrtqdrt

rqdrtlog2

loglog1pexp2

expexpm1

atan2atanasinacos

sincostan

0 5 10 15 20 25 30







Acknowledgments


References

























1 Introduction








[119865 (119905)]119896

= sum119899gt0

119865 (119899 119896) 119905119899

(1)





119860Δ

(119899 119896) = 120572119896

119865Δ

(119899 119896) (2)



119860Δ

(119899 119896) = 120572119899

119865Δ

(119899 119896) (3)


119891(119899)119905119899 119877(119905) = sum119899gt0

119903(119899)119905119899


119860Δ

(119899 119896) =

119899

sum119898=119896

119865Δ

(119899119898)119866Δ

(119898 119896) (4)


119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899



119886(119899)119905119899

are

119886 (119899) =

119899

sum119896=1

119865Δ

(119899 119896) 119903 (119896)

119886 (0) = 119903 (0)

(5)




119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899


119899ge0119886(119899)119905119899 are

119886 (119899) =

119899

sum119896=0

119891 (119896) 119903 (119899 minus 119896) (6)




119910119899(119909) =

119899

sum119896=0

(119899 + 119896)

(119899 minus 119896)119896(119909

2)119896

(7)


119901119899(119909) = 119909

119899

119910119899minus1

(1

119909) (8)


119901119899(119909) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)(1

2)119899minus119896

119909119896 (9)


sum119899=0

119901119899(119909)

119899119905119899


(10)




119862Δ

(119899 119896) =119896

119899(2119899 minus 119896 minus 1

119899 minus 1) (11)



2)

119861 (119905) = 119890119905

=

infin

sum119899=0

1

119899119905119899

(12)


119860Δ

(119899 119896) = (2119909)119896

(1

2)119899

119862Δ

(119899 119896) = 2119896minus119899

119909119896119896

119899(2119899 minus 119896 minus 1

119899 minus 1)

(13)


119899=0119887(119899)119905119899

are equal to

119887 (119899) =1

119899 (14)


119901119899(119909) = 119899

119899

sum119896=1

119860Δ

(119899 119896) 119887 (119896) =

119899

sum119896=1

(2119899 minus 119896 minus 1)


119909119896

(15)




119898119899+1

(119909 120573 119888) =(119888 minus 1)

119899

119888119899

sdot ((119909 minus 119887119899)119898119899(119909 120573 119888) minus 120582

119899119898119899minus1

(119909 120573 119888))

(16)

where

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) = 119909 minus 119887

0

119887119899=

(1 + 119888) 119899 + 120573119888

1 minus 119888

120582119899=

119888119899 (119899 + 120573 minus 1)

(1 minus 119888)2

(17)


1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) =

119909 (119888 minus 1) + 120573119888

119888

1198982(119909 120573 119888)

=1199092 (119888 minus 1)

2

+ 119909 ((2120573 + 1) 1198882 minus 2120573119888 minus 1) + (120573 + 1) 1205731198882

1198882

(18)


infin

sum119899=0

119898119899(119909 120573 119888)

119899119905119899

= (1 minus119905

119888)119909


(19)





119860 (119905) = (1 minus119905

119888)119909

=

infin

sum119899=0

(119909

119899) (minus1)

119899

119888minus119899

119905119899


=

infin

sum119899=0


119899) (minus1)

119899

119905119899

(20)


suminfin

119899=0119886(119899)119905119899 and 119861(119905) = sum

infin


follows

119886 (119899) = (119909

119899) (minus1)

119899

119888minus119899

119887 (119899) = (minus119909 minus 120573

119899) (minus1)

119899

(21)


119898119899(119909 120573 119888) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896)

= (minus1)119899

119899

119899

sum119896=0

(119909

119896)(


119899 minus 119896) 119888minus119896

(22)



119872119899+1

(119909 120575 120578) = (119909 minus 119887119899)119872119899(119909 120575 120578) minus 120582

119899119872119899minus1

(119909 120575 120578)

(23)

where1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 119887

0

119887119899= (2119899 + 120578) 120575

120582119899= (1205752

+ 1) 119899 (119899 + 120578 minus 1)

(24)


1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 120575120578

1198722(119909 120575 120578) = 119909

2

minus 1199092120575 (1 + 120578) + 120578 ((120578 + 1) 1205752

minus 1)

(25)


infin

sum119899=0

119872119899(119909 120575 120578)

119899119905119899

= ((1 + 120575119905)2

+ 1199052

)minus1205782


1 + 120575119905))

(26)




119860 (119905) = ((1 + 120575119905)2

+ 1199052

)minus1205782

= (1 + 2120575119905 + (1205752

+ 1) 1199052

)minus1205782

119861 (119905) = exp (119909tanminus1 ( 119905

1 + 120575119905))

(27)


1(1198602(119905)) and we expand 119860


theorem

1198601(119905) = (1 + 119905)

minus1205782

=

infin

sum119899=0

(minus120578

2119899

) 119905119899

1198602(119905) = 2120575119905 + (120575

2

+ 1) 1199052

(28)


infin

119899=01198861(119899)119905119899

are

1198861(119899) = (

minus120578

2119899

) (29)


1198862119896minus119899

119887119899minus119896

(119896

119899 minus 119896) (30)


119860Δ

2(119899 119896) = (2120575)

2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896) (31)


infin

119899=0119886(119899)119905119899

119886 (0) = 1198861(0)

119886 (119899) =

119899

sum119896=1

119860Δ

2(119899 119896) 119886

1(119896)

(32)


119886 (0) = 1

119886 (119899) =

119899

sum119896=1

(2120575)2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896)(

minus120578

2119896

) (33)


1(1198612(119905)) where

1198611(119905) = 119890

119905

=

infin

sum119899=0

1

119899119905119899

1198612(119905) = 119909tanminus1 ( 119905

1 + 120575119905)

(34)



functions 1199091198621(1198622(119905)) where

1198621(119905) = tanminus1 (119905)

1198622(119905) =

119905

1 + 120575119905

(35)


((minus1)(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896] (36)

where [119895



119862Δ

1(119899 119896) = ((minus1)

(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

sdot

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896]

(37)


(119899 minus 1

119896 minus 1) 119886119896

119887119899minus119896

(38)


to

119862Δ

2(119899 119896) = (

119899 minus 1

119896 minus 1) (minus120575)

119899minus119896

(39)


2(119905)

119861Δ

2(119899 119896) = 119909

119896

119899

sum119898=119896

119862Δ

2(119899119898) 119862

Δ

1(119898 119896) (40)


infin

119899=01198871(119899)119905119899

are defined by

1198871(119899) =

1

119899 (41)


infin

119899=0119887(119899)119905119899

119887 (0) = 1198871(0)

119887 (119899) =

119899

sum119896=1

119861Δ

2(119899 119896) 119887

1(119896)

(42)


119887 (0) = 1

119887 (119899) =

119899

sum119896=1

119909119896

119899

119899

sum119898=119896

(119899

119898)120575119899minus119898

(1 + (minus1)

119898+119896

(minus1)(119898+119896)2minus119899

)

sdot

119898

sum119895=119896


(119895 minus 1)(119898

119895)[

119895

119896]

(43)


119872119899(119909 120575 120578) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896) (44)



Acknowledgment


References





































119892 (119911) =1

119911+ 1198870+

infin

sum119899=1

119887119899119911119899 (1)




1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911) (2)






119899| le

(119861 minus 119860)(119899 + 1) for 119892 isin Σ[119860 119861]



ℎ (119908) = 119892minus1

(119908) =1

119908+

infin

sum119899=0

119861119899119908119899

=1


1

119899119870119899

119899+1(1198870 1198871 119887

119899) 119908119899

(3)

where 119908 isin D

119870119899

119899+1(1198870 1198871 119887

119899)

= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)


3119887119899minus4

0(1198874+ 311988711198872)

+ sum119895ge5

119887119899minus119895

0119881119895

(4)


variables 1198871 1198872 119887

119899





119899|



119899| and




119899=1119901119899119911119899 be so that 119877119890(119901(119911)) gt 0


100381610038161003816100381610038161199012+ 1205721199012

1

10038161003816100381610038161003816le 2 + 120572

1003816100381610038161003816119901110038161003816100381610038162

(5)



119899=1119888119899119911119899


100381610038161003816100381610038161198882+ 1205741198882

1

10038161003816100381610038161003816le 1 + (120574 minus 1)

1003816100381610038161003816119888110038161003816100381610038162

(6)

Proof Write

119901 (119911) =[1 + 120593 (119911)]

[1 minus 120593 (119911)] (7)

where 119901(119911) = 1+suminfin



1= 21198881and 119901

2=

2(1198882+ 11988821)


2= 2(1198882+ 11988821) in (5) we

obtain100381610038161003816100381610038162 (1198882+ 1198882

1) + 120572 (2119888

1)210038161003816100381610038161003816le 2 + 120572

10038161003816100381610038162119888110038161003816100381610038162 (8)

or100381610038161003816100381610038161198882+ (1 + 2120572) 119888

2

1

10038161003816100381610038161003816le 1 + 2120572

1003816100381610038161003816119888110038161003816100381610038162

(9)




(119894)10038161003816100381610038161198870

1003816100381610038161003816 le

119861 minus 119860


119861 minus 119860 119900119905ℎ119890119903119908119894119904119890

(119894119894)10038161003816100381610038161198871

1003816100381610038161003816

le

119861 minus 119860


2 (119861 minus 119860)|1198870|2

119894119891 0 le 119860 le 1 minus 2119861

119861 minus 119860

2 119900119905ℎ119890119903119908119894119904119890

(119894119894119894)100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2

(119894V) 10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 119894119891 119887

119896= 0 119891119900119903 0 le 119896 le 119899 minus 1

(10)


1199111198921015840 (119911)

119892 (119911)= minus1 minus

infin

sum119899=0

119865119899+1

(1198870 1198871 1198872 119887

119899) 119911119899+1

(11)

where 119865119899+1

(1198870 1198871 1198872 119887




5=



119865119899+1

(1198870 1198871 119887

119899)


119899+1=119899+1

119860 (1198941 1198942 119894

119899+1) 1198871198941

01198871198942

1sdot sdot sdot 119887119894119899+1

119899

(12)

where

119860 (1198941 1198942 119894

119899+1)

= (minus1)(119899+1)+2119894

1+sdotsdotsdot+(119899+2)119894

119899+1

(1198941+ 1198942+ sdot sdot sdot + 119894

119899+1minus 1) (119899 + 1)

(1198941) (1198942) sdot sdot sdot (119894

119899+1)

(13)



119908ℎ1015840 (119908)


infin

sum119899=1

119865119899+1

(1198610 1198611 1198612 119861

119899) 119908119899+1

(14)

where119865119899+1

(1198610 1198611 1198612 119861


119899 + 1 given by

119865119899+1

= minus119899 (119899 minus (119899 + 1))

119899 (119899 minus 2119899)119861119899

0

minus119899 (119899 minus (119899 + 1))


01198611

minus119899 (119899 minus (119899 + 1))


01198612

minus119899 (119899 minus (119899 + 1))


0

sdot (1198613+119899 minus (2119899 minus 3)

21198612


119861119899minus119895

0119870119895

(15)


variables 1198611 1198612 119861

119899minus1and

1198610= minus1198870 119861

119899= minus

1

119899119870119899

119899+1(1198870 1198871 119887

119899) (16)


1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198881 1198882 119888

119899 119861) 119911119899

(17)

119908ℎ1015840 (119908)

ℎ (119908)= minus

1 + 119860120595 (119908)

1 + 119861120595 (119908)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198891 1198892 119889

119899 119861) 119908

119899

(18)


119899=1119888119899119911119899 and 120595(119908) = sum

infin

119899=1119889119899119908119899 are two



119899(1198961 1198962 119896

119899 119861) are given by

119870119901

119899(1198961 1198962 119896

119899 119861)

=119901

(119901 minus 119899)119899119896119899

1119861119899minus1

+119901


11198962119861119899minus2

+119901


11198963119861119899minus3

+119901


1

sdot [1198964119861119899minus4

+119901 minus 119899 + 3

21198962

2119861]

+119901


1

sdot [1198965119861119899minus5

+ (119901 minus 119899 + 4) 11989621198963119861]

+ sum119895ge6

119896119899minus119895

1119883119895

(19)


variables 1198962 1198963 119896

119899


minus119865119899+1

(1198870 1198871 1198872 119887

119899) = (119860 minus 119861)119870

minus1

119899+1(1198881 1198882 119888

119899+1 119861)

(20)


minus 119865119899+1

(1198610 1198611 1198612 119861

119899)

= (119860 minus 119861)119870minus1

119899+1(1198891 1198892 119889

119899+1 119861)

(21)


1198870= (119860 minus 119861) 119888

1

minus1198870= (119860 minus 119861) 119889

1

(22)

21198871minus 1198872

0= (119861 minus 119860) (119888

2

1119861 minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (119889

2

1119861 minus 1198892)

(23)




0and then adding

them gives

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus 1198611198882

1minus 1198611198892

1) (24)



21198872

0= (119861 minus 119860) (119888

2+ 1198892minus

2119861

(119861 minus 119860)21198872

0) (25)


sides we obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(119861 minus 119860)

2

(1198882+ 1198892)

2 (2119861 minus 119860)le

119861 minus 119860

radic2119861 minus 119860 (26)



119861 minus 119860



we obtain

41198871= (119861 minus 119860) [(119889

2minus 1198882) minus (119889

2

1minus 1198882

1) 119861] = (119861 minus 119860) (119889

2minus 1198882)

(28)


100381610038161003816100381611988711003816100381610038161003816 le

119861 minus 119860

2 (29)


21198871= (119861 minus 119860) (119889

2+ 1198601198892

1minus 2119861119889

2

1) (30)


100381610038161003816100381611988711003816100381610038161003816 le

1

2(119861 minus 119860) (

100381610038161003816100381610038161198892+ 1198601198892

1

10038161003816100381610038161003816+ 2119861

1003816100381610038161003816119889110038161003816100381610038162

)

le1

2(119861 minus 119860) (1 + (119860 minus 1)

1003816100381610038161003816119889110038161003816100381610038162

+ 211986110038161003816100381610038161198891

10038161003816100381610038162

)

=119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

(31)




2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

lt119861 minus 119860

2 (32)


21198871minus 1198872

0= (119861 minus 119860) (

119861

(119861 minus 119860)21198872

0minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (

119861

(119860 minus 119861)21198872

0minus 1198892)

(33)


100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2 (34)


(119899 + 1) 119887119899= (119860 minus 119861) 119888

119899+1 (35)


obtain

10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 (36)







0|2 given by






References



























1 Introductions


119891 (119911) = 119911 +

infin

sum120581=2

119886120581119911120581

(1)


119867[119886 119899] = 119901 isin 119867 119901 (119911) = 119886 + 119886119899119911119899

+119886119899+1

119911119899+1


(2)


119865 (119911) = 119911120573

+

infin

sum120581=2

120582120581119886120581119911120573+120581minus1

120573 isin N

120582120581= 120573120581minus1

119911 =1

120573 120573 |119911| lt 1

(3)


(119891 lowast 119892) (119911) =

infin

sum119896=0

119886119896119887119896119911119896

(4)


119896=0119886119896119911119896 and 119892(119911) = sum

infin

119896=0119887119896119911119896 are analytic in

119880

Let 120595(119911) = 119911120573 + sum

infin



120595 (119911) lowast119911120573

1 minus 120573119911 119911 =

1

120573 120573 isin N 120573 |119911| lt 1 (5)



119876(120485) (120572 120591 120574 120573)

= 119865 (119911) isin 119867 (120573) R119890

times120572[119865(119911)]

(120485)

[119911120573](120485)

+120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

gt 120574 119911 isin 119880

(6)

where

119911 = 0 120573 gt 120485 120485 isin N cup 0 also 120572 + 120591 = 120574 (7)







119891 (119911) =1

120572 + 120591int|119909|


times

infin

sum120581=0

(120572 + 120591) 119909120581

119911120581+1

(120581 + 1) 120591 + 120572] 119889120583 (119909)

(8)




Θ120572120573

(119911) = (120573int119911

0

119905120573minus120572minus1

[119891 (119905)]120572

119892 (119905) 119889119905)

1120573

(10)






119901 (119911) = int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909) (|119911| lt 1) (11)



119892 (119911) +1199111198921015840 (119911)

119888≺ ℎ (119911) (12)

then

119892 (119911) ≺ 119902 (119911) =119888119911minus119888120581

120581int119911

0

119905(119888minus120581)120581

ℎ (119905) 119889119905 ≺ ℎ (119911) (13)



1199111199011015840 (119911)

119901 (119911)≺ ℎ (119911) (14)

then

119901 (119911) ≺ 119902 (119911) = 119886 exp [1

119899int119911

0

ℎ (120585) 120585minus1

119889120585] (15)




(ii) |1199011015840(119911)| le 2R119890119901(119911)(1 minus 1199032)


100381610038161003816100381610038161003816100381610038161003816

1199111199011015840 (119911)

119901 (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 119901 (119911)

(1 minus 119903)2

(16)




(120485)

(120572120591120574120573)




119911120485

[119865 (119911)](120485)

=

120573

119875120485

(120572 + 120591)int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120573 minus 120485) (120572 + 120591) 119909120581119911120573+120581

(120572 + 120591) (120573 minus 120485) + 120581120591] 119889120583 (119909)

(17)

where 119899119875119903




119901 (119911)

=

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

isin 119867 [1 119899]

(18)

ByTheorem 3

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

= int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909)

(19)


[119865 (119911)](120485+1)

[119911120573](120485+1)

+120572[119865 (119911)]

(120485)

120591[119911120573](120485)

=1

120591int|119909|=1

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909119911

1 minus 119909119911119889120583 (119909)

(20)

which yields

119911((120572+120591)(120485minus120573)+120591)120591

int119911

0

[[119865 (120585)]

(120485+1)

[120585120573](120485+1)

+120572[119865 (120585)]

(120485)

120591[120585120573](120485)

]


119889120585

=1

120591int|119909|=1

119911((120572+120591)(120485minus120573)+120591)120591

times int119911

0

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909120585

1 minus 119909120585


119889120585 119889120583 (119909)

(21)



shows that 119865(119911) isin 119876(120485)(120572 120591 120574 120573)



119865 (119911) =1

120572 + 120591int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581119911120573+120581

120572120573 + 120591 (120581 + 120573)] 119889120583 (119909)

(22)


1




10038161003816100381610038161198861205811003816100381610038161003816 le 119867 (120572 120591 120574 120573) (23)

where

119867(120572 120591 120574 120573) =2 (120572 + 120591 minus 120574) (120573 minus 120485) [

120573

119875120485]

120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(24)


119911120485

[119865119909(119911)](120485)

120573119875120485

= 119911120573

+

infin

sum120581=2


(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)(|119909| = 1)

(25)




119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [[120581 minus 1] 119903

120581minus1]

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

le2119903R119890 Ψ (119911)

(1 minus 119903)2

(26)


119866 (119911) equiv120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

+120572[119865(119911)]

(120485)

[119911120573](120485)

minus 120574 (27)

and then119911Ψ1015840 (119911)

Ψ (119911)=

suminfin

120581=2120599 (120572 120591 120574 120573) [120581 minus 1] 119886

120581119911120581minus1

1 + suminfin

120581=2120599 (120572 120591 120574 120573) 119886

120581119911120581minus1

(28)

where

120599 (120572 120591 120574 120573) =120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591 minus 120574) [120573

119875(120485)

]

(29)

From (23) and (28) we got100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [120581 minus 1] 119903

120581minus1

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

(120581 ge 2) (30)


119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 Ψ (119911)

(1 minus 119903)2

(31)

Sinceinfin

sum120581=2


lt 2119903R119890 Ψ (119911)

1 +

infin

sum120581=2

2 (120573 minus 120485) 119903120581minus1

gt (1 minus 119903)2

(32)




(120485)

(120572 120591 120574 120573)



1199111198661015840 (119911)

119866 (119911) ≺ ℎ (119911) (33)

then

119866 (119911) ≺ 119902 (119911) = (120572 + 120591 minus 120574) exp [119899minus1

int119911

0

ℎ (120585) 120585minus1

119889120585 ] (34)



119866 (119911) = (120572 + 120591 minus 120574) +

infin

sum120581=2

120582120581[ [120573+120581minus1]119875

120485+1] 119886120581119911120581minus1

120573119875120485+1

(35)


1199111198661015840

(119911)

119866 (119911) ≺ ℎ (119911) (36)




1199111198661015840 (119911)

119866 (119911) ≺

radic119911

1 minus 119911(37)

then

119866 (119911) ≺ (120572 + 120591 minus 120574) 1 + radic119911

1 minus radic119911

1119899

120572 + 120591 = 120574 (38)




120573 gt 120485 + 1 with 120572 + 120591 = 0 If

119866 (119911) ≺ (120572 + 120591)1 + 119860119911

1 + 119861119911 (39)

then[119865(119911)]

(120485)

[119911120573](120485)

≺(120572 + 120591) (120573 minus 120485)

120591120581119911(120572+120591)(120485minus120573)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(40)


1

(120572 + 120591)119866 (119911) = 1 +

infin

sum120581=2

119862120581119886120581119911120581minus1

(41)

where

119862120581=

120582120581[ (120573+120581minus1)119875

120485+1] sdot [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591) [120573

119875120485+1

] (42)

Let 119892(119911) = [119865(119911)](120485)

[119911120573](120485) then 119892(119911) = 1 + sum

infin

120581=2120603120581119886120581119911120581minus1

where 120603120581= 120582120581[(120573+120581minus1)

119875120485+1

][(120573)

119875120485] and from (39) we have

119892 (119911) +120591

(120572 + 120591) (120573 minus 120485)1199111198921015840

(119911) ≺ (1 + 119860119911

1 + 119861119911) = ℎ (119911) (43)


119892 (119911) ≺(120572 + 120591) (120573 minus 120485)

120591120581119911minus(120572+120591)(120573minus120485)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(44)


Corollary 16 Let

119876(0)

(120572 120591 0 120573) =120572 (119865 (119911))

119911120573+

120591[119865 (119911)]1015840

[119911120573]1015840

(45)

If

119876(0)

(120572 120591 0 120573) ≺ (120572 + 120591) (1 + 119860119911

1 + 119861119911) (46)

then

119865 (119911)

119911120573≺

120572120573 + 120591 (120573 minus 120581)

120591120581119911minus120573(120572+120591)120591120581

times int119911

0

120585(120573(120572+120591)minus120591120581)120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(47)






Acknowledgments



References





















1 Introduction


119891 (119911) =119892 (119911)

ℎ (119911) (1)






R ℎ (119911) gt 0 (119911 isin 119880lowast

) (2)


119891 ≺ 119892 in 119880 or 119891 (119911) ≺ 119892 (119911) (119911 isin 119880lowast

) (3)


|119908 (119911)| le |119911| 119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880lowast

) (4)


119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0)

119891 (119880) sube 119892 (119880) (119911 isin 119880lowast

) (5)





2 Preliminaries


119891 (119911) =1

119911+

infin

sum119899=1

119886119899119911119899

(6)




119891119895(119911) =

1

119911+

infin

sum119899=1

119886119899119895119911119899

(7)


and 1198912(119911) by

(1198911lowast 1198912) =

1

119911+

infin

sum119899=1

11988611989911198861198992119911119899

(8)


119902120582120583

(119911) =1

119911+

infin

sum119899=1

(120582

119899 + 1 + 120582)

120583

119911119899

(120582 gt 0 120583 ge 0) (9)



119871120582120583119891 (119911) = (119891 (119911) lowast 119902

120582120583(119911))

=1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(10)


119905119886119892 and 119871


tively by

119871120582120583119892 (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198871198991003816100381610038161003816 119911119899


119871120582120583ℎ (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198881198991003816100381610038161003816 119911119899


ℎ (119911) isin Σ)

(11)


119871120582120583

(119891 lowast 119892) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast)

119871120582120583

(119891 lowast ℎ) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198881198991003816100381610038161003816 119911119899


)

(12)






property

120588119871120582120583

(119891 lowast 119892) (119911)

119871120582120583

(119891 lowast ℎ) (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911 (119911 isin 119880

lowast

) (13)


and 119871(120582 120583)(119891 lowast ℎ)(119911) = 0


120601 (120572 120573 119911) =1

119911+

infin

sum119899=0

(120572)119899+1

(120573)119899+1

119911119899

(14)



120601 (120572 120573 119911) =1

119911 21198651(1 120572 120573 119911) (15)

where

21198651(119887 120572 120573 119911) =

infin

sum119899=0

(119887)119899(120572)119899

(120573)119899

119911119899

119899(16)


120582120583(119911)


119871 (120572 120573 120582 120583) 119891 (119911) = (119891 (119911) lowast 120601 (120572 120573 119911) lowast 119902120582120583

(119911))

=1

119911+

infin

sum119899=1

(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(17)




119911(119871 (120572 120573 120582 120583) 119891 (119911))1015840

= 120572119871 (120572 + 1 120573 120582 120583) 119891 (119911)

minus (120572 + 1) 119871 (120572 120573 120582 120583) 119891 (119911) (18)


120588119871 (120572 + 1 120573 120582 120583) 119891 (119911)

119871 (120572 120573 120582 120583) 119891 (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911

(0 le 119861 lt 119860 le 1 120588 gt 0)

(19)



relation (19) above






infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861)

minus10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(20)


119891119899(119911) =

1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

(21)



1003816100381610038161003816100381610038161003816100381610038161003816

120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899) 119911119899+1

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899119911119899+1

)

minus11003816100381610038161003816100381610038161003816100381610038161003816

le 120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899)10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899

10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816)

minus1

le 1

(22)





100381610038161003816100381610038161003816100381610038161003816lt 1

(119911 isin 119880lowast)

(23)


Σ120583




120582(120588 119860 119861) then

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119899 ge 1

(24)


(21)


120582(120588 119860 119861)



1

119903minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903

le1003816100381610038161003816119891 (119911)

1003816100381610038161003816

le1

119903+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903


1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

le100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(25)



inequalityinfin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(26)


1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

le1

119903+ 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

119903+ 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|minus

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

ge1

119903minus 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

ge1

119903minus 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(27)


sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(119860 minus 119861) (2 + 120582)

120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(28)

Hence

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2+

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

le1

1199032+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

ge1

1199032minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816

ge1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(29)



120582(120588 119860 119861) which




disk |119911| lt 1199031 where

1199031= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(30)



119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (31)

For |119911| lt 1199031 we have

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 1 + 120582))

120583

119886119899119911119899

1119911 + suminfin

119899=1(120582(119899 + 1 + 120582))

120583

119886119899119911119899

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

100381610038161003816100381610038161003816100381610038161003816

lesuminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(32)


sum119899=1

(119899 + 1) (120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1


infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

)

(33)

or

suminfin

119899=1(119899 + 2 minus 120575) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(1 minus 120575)le 1 (34)


(119899 + 2 minus 120575) (120582(119899 + 1 + 120582))120583

|119911|119899+1

(1 minus 120575)

le (120582

119899 + 120582 + 1)

120583

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899 ge 1

(35)



|119911| le (1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899+1

(36)




disk |119911| lt 1199032 where

1199032= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(37)



1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891 (119911))10158401015840

(119891 (119911))1015840+ 2

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (38)






infin

sum119899=1

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(39)


119891119899(119911)

=1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

119899 ge 1

(40)




120582(120588 120572 120573 119860 119861) then1003816100381610038161003816119886119899

1003816100381610038161003816

le(119860 minus 119861) (119899 + 120582 + 1)

120583

(120573)119899+1

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861)) (120572)

119899+1

119899 ge 1

(41)


(40)




1199033= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(42)





1199034= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(43)






Acknowledgment


References






















1 Introduction


Φ (119911 119904 119886) =

infin

sum119899=0

119911119899

(119899 + 119886)119904


0 119904 isin C when |119911| lt 1

R (119904) gt 1 when |119911| = 1)

(1)


119904(120585) defined by

120577 (119904) =

infin

sum119899=1

1

119899119904= Φ (1 119904 1) = 120577 (119904 1) (R (119904) gt 1)

120577 (119904 119886) =

infin

sum119899=0

1

(119899 + 119886)119904


0)

ℓ119904(120585) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 1)119904= Φ (e2120587119894120585 119904 1) (R (119904) gt 1 120585 isin R)

(2)


119904(119911)

Li119904(119911) =

infin

sum119899=1

119911119899

119899119904= 119911Φ (119911 119904 1)


(3)


120601 (120585 119886 119904) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 119886)119904= Φ (e2120587119894120585 119904 119886)



when 120585 isin Z)

(4)




Φ (119911 119904 119886) =1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

1 minus 119911eminus119905d119905

(R (119886) gt 0R (119904) gt 0 when |119911| ≦ 1 (119911 = 1) R (119904) gt 1

when 119911 = 1) (5)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) =

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

119911119899

(119899 + 119886)119904

(119901 119902 isin N0 120582

119895isin C (119895 = 1 119901)

119886 120583119895isin C Z

minus

0(119895 = 1 119902)

120588119895 120590

119896isin R

+

(119895 = 1 119901 119896 = 1 119902)





lowast

)

(6)

with

nablalowast

= (

119901

prod119895=1

120588minus120588119895

119895) sdot (

119902

prod119895=1

120590120590119895

119895)

Δ =

119902

sum119895=1

120590119895minus

119901

sum119895=1

120588119895

Ξ = 119904 +

119902

sum119895=1

120583119895minus

119901

sum119895=1

120582119895+119901 minus 119902

2

(7)



(120582)120581=Γ (120582 + 120581)

Γ (120582)

=


1 (120581 = 0 120582 isin C 0)

(8)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0)

(9)



119898or

119897Ψ119898(119897 119898 isin N

0) involved


119897119865119898(119897 119898 isin N




1198871 119887

119898such that

119886119895isin C (119895 = 1 119897) 119887

119895isin C Z

minus

0(119895 = 1 119898)

(10)


119897Ψlowast

119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

=

infin

sum119899=0

(1198861)1198601119899sdot sdot sdot (119886

119897)119860119897119899

(1198871)1198611119899sdot sdot sdot (119887

119898)119861119898119899

119911119899

119899

=Γ (119887


119898)

Γ (1198861) sdot sdot sdot Γ (119886

119897) 119897Ψ119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

(119860119895gt 0 (119895 = 1 119897) 119861

119895gt 0 (119895 = 1 119898)

1 +

119898

sum119895=1

119861119895minus

119897

sum119895=1

119860119895≧ 0)

(11)


|119911| lt nabla = (

119897

prod119895=1

119860minus119860119895

119895) sdot (

119898

prod119895=1

119861119861119895

119895) (12)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

Γ (119904)intinfin

0


119905120582)

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0)

(13)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 0 120582)

= Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

= e119887Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 0)

(14)


119901 minus 1 = 119902 = 0 (1205821= 120583 120588

1= 1) (15)


Θ120582

120583(119911 119904 119886 119887)

=1

Γ (119904)intinfin

0


119905120582) (1 minus 119911eminus119905)

minus120583

d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0 120583 isin C)

(16)


R (119904) gt 0 when 119887 = 0 |119911| ≦ 1 (119911 = 1) (17)

or

R (119904 minus 120583) gt 0 when 119887 = 0 119911 = 1 (18)




Θ120582


Θ120582

120583(119911 119904 119886 0) = Φ

lowast

120583(119911 119904 119886) =

1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

(1 minus 119911eminus119905)120583d119905

(R (119886) gt 0 R (119904) gt 0 when |119911| ≦ 1 (119911 = 1)

R (119904 minus 120583) gt 0 when 119911 = 1)

(19)



[4] as follows

Φlowast

120583(119911 119904 119886) =

infin

sum119899=0

(120583)119899

(119886 + 119899)119904

119911119899

119899 (20)


120583(119911 119904 119886 119887) that is


Θ1

1(1 119904 119886 119887)

= 120577119887(119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1


119905) d119905

(21)




(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899

119899(120582 gt 0)

(22)



119867119898119899

pq (119911)

= 119867119898119899

pq [119911 |(119886

1 119860

1) (119886p 119860p)

(1198871 119861

1) (119887q 119861q)

]

=1

2120587119894intL

Ξ (119904) 119911minus119904d119904 (119911 isin C 0

1003816100381610038161003816arg (119911)1003816100381610038161003816 lt 120587)

(23)

where

Ξ (119904) =prod

119898

119895=1Γ (119887

119895+ 119861

119895119904) sdot prod

119899

119895=1Γ (1 minus 119886

119895minus 119860

119895119904)

prodp

119895=119899+1Γ (119886

119895+ 119860

119895119904) sdot prod

q

119895=119898+1Γ (1 minus 119887

119895minus 119861

119895119904)

(24)


119895gt 0 (119895 =

1 p) 119861119895gt 0 (119895 = 1 q) 119886

119895isin C (119895 = 1 p)



Γ (119887119895+ 119861

119895119904)

119898

119895=1

(25)


Γ (1 minus 119886119895+ 119860

119895119904)

119899

119895=1

(26)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902





1 minus 119905)

=

infin

sum119899=0

119871(120572)

119899(119909) 119905

119899

(|119905| lt 1 120572 isin C)

(27)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=eminus119887

Γ (119904)

infin

sum119899=0

119899

sum119896=0

(minus1)119896

(119899

119896) sdot Γ (119904 + 120582 (120572 + 119896 + 1))

sdot 119871(120572)

119899(119887)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 + 120582 (120572 + 119895 + 1) 119886)

(R (119886) gt 0 R (119904 + 120582120572) gt minus120582)

(28)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) (29)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



D120572

119911119911

119901

119891 (119911) =1

Γ (minus120572)int119911

0

119891 (120585) 120585119901


(R (120572) lt 0 R (119901) gt 1)

(30)


D120572

119911119911

119901

119891 (119911) =d119898

d119911119898D

120572minus119898

119911119911

119901

119891 (119911) (31)



Im(120585)

Branch line for

Branch line for


exp[p(ln(g(120585)))]

C1

C2

C3

C4

Re(120585)

z

a

gminus1(0)





119901


119863120572

119892(119911)[119892 (119911)]

119901

119891 (119911)

=eminus119894120587119901Γ (1 + 120572)4120587 sin (120587119901)

times int119862(119911+119892

minus1(0)+119911minus119892

minus1(0)minus119865(119886)119865(119886))

sdot119891 (120585) [119892 (120585)]

119901

1198921015840 (120585)

[119892 (120585) minus 119892 (119911)]120572+1

d120585

(32)



1cup119862

2cup119862

3cup119862








119892(119911)[119892(119911)]

119901



119863120572

119911119911

119901

=Γ (1 + 119901)

Γ (1 + 119901 minus 120572)119911119901minus120572

(R (119901) gt minus1) (33)



11205881199011205901120590119902)

12058211205821199011205831120583119902


119863120572

119911119911

120573minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899)

1198871120582

| (119904 1) (01

120582)]119863

120572

119911119911

120573minus1119911119899

119899

=Γ (120573)

120582Γ (119904) Γ (120573 minus 120572)

infin

sum119899=0

(120573)119899

(120573 minus 120572)119899

sdot

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899+120573minus120572minus1

119899

=Γ (120573)


Φ(1205881120588119901112059011205901199021)

12058211205821199011205731205831120583119902120573minus120572

(119911 119904 119886 119887 120582)







119863120572

119892(119911)119891 (119911) = 119863

120572

ℎ(119911)119891 (119911) 119892

1015840(119911)

ℎ1015840(119911)(ℎ(119908) minus ℎ(119911)

119892(119908) minus 119892(119911))

120572+1

100381610038161003816100381610038161003816100381610038161003816119908=119911

(35)




119911119901119874120572

120573119891 (119911)

= 119901(119911119901minus1

)minus120572

sdot119911

119874120572

120573119891 (119911) (119911

119901minus1

)120572

119901minus1

prod119904=1

(1 minus119911

119908eminus(2120587119894119904)119901)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(36)

where

119892(119911)119874120572


Γ (120573)

Γ (120572)[119892 (119911)]

1minus120573

119863120572minus120573

119892(119911)[119892 (119911)]

120572minus1

sdot sdot sdot

(37)



120579 (119911) = (119911 minus 1199110) 119902 (119911) (38)


119870

=0 1 [119888] ([119888] 119905ℎ119890 119897119886119903119892119890119904119905 119894119899119905119890119892119890119903 119899119900119905 119892119903119890119886119905119890119903 119905ℎ119886119899 119888)

(39)


0and let

120596 = exp (2120587119894119886) (40)



sum119896isin119870

119888minus1

120596minus120574119896

119891 (120579minus1

(120579 (119911) 120596119896

))

=

infin


[120579 (119911)]119888119899+120574

Γ (119888119899 + 120574 + 1)

sdot119863119888119899+120574

119911minus119887119891 (119911) 120579

1015840

(119911) (119911minus119911

0

120579 (119911))

119888119899+120574+1

100381610038161003816100381610038161003816100381610038161003816119911=1199110

(1003816100381610038161003816119911minus1199110

1003816100381610038161003816=100381610038161003816100381611991101003816100381610038161003816)

(41)



119891 (119911) = 119888

infin


(119911 minus 1199110)119888119899+120574

Γ(119888119899 + 120574 + 1)119863

119888119899+120574

119911minus119887119891(119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199110

(42)



1)(119911minus119911

2)) where 119911

1and 119911

2are two



120596 = exp(2120587119894119886) (43)


1and 119911


curves

119862 (119905) 119862 (119905) subR 0 lt 119905 ≦ 119903 (44)

be defined by

119862 (119905) = 1198621(119905) cup 119862

2(119905)

= 119911 1003816100381610038161003816120582119905 (1199111 1199112 119911)

1003816100381610038161003816 =1003816100381610038161003816100381610038161003816120582119905(119911

1 119911

21199111+ 119911

2

2)1003816100381610038161003816100381610038161003816

(45)

where

120582119905(119911

1 119911

2 119911) = [119911 minus

1199111+ 119911

2

2+ 119905 (

1199111minus 119911

2

2)]

sdot [119911 minus (1199111+ 119911

2

2) minus 119905 (

1199111minus 119911

2

2)]

(46)


1+ 119911



1199111+ 119911

2

2+ (

1199111minus 119911

2

2) 119905 (47)


2

2minus (

1199111minus 119911

2

2) 119905 (48)

Im(120585)

Branch line for

Branch line for

C1

C2

z1

z2

Re(120585)

z

c =z1 + z2

2

(120585 minus z2)120583120579(120585)

aminus120574minus1

(120585 minus z1)120579(120585)

aminus120574minus1



[(119911 minus 1199111) (119911 minus 119911

2)]

120582


2))))

(49)


plusmn

defined by

119871plusmn=

119911 119911 =1199111+ 119911

2

2plusmn 119905 (

1199111minus 119911

2

2) (0 ≦ 119905 ≦ 1)

119911 119911 =1199111+ 119911

2

2plusmn 119894119905 (

1199111minus 119911

2

2) (119905 lt 0)

(50)



1)(119911minus119911

2) gt 0 (v)


2)119901


+cup 119871


119911minus1199112

(119911 minus 1199112)119901


119870 = 119896 119896 isin N arg(120582119905(119911

1 119911

21199111+ 119911

2

2))

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) +

2120587119896

119886

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) + 2120587

(51)


defined by

120585 =1199111+ 119911

2

2+1199111minus 119911

2


sum119896isin119870

119888minus1120596minus120574119896

1199111minus 119911

2

119891 (120601minus1

(120596119896

120601 (119911)))

times [120601minus1

(120596119896

120601(119911)) minus 1199111]][120601

minus1

(120596119896

120601(119911)) minus 1199112]120583

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

[120601 (119911)]119888119899+120574

(52)


where

120601 (119911) =119911 minus 119911

1

119911 minus 1199112

(53)



2)120583

1199111minus 119911

2

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

(119911 minus 119911

1

119911 minus 1199112

)

119888119899+120574

(54)


2) where 119911

1and 119911




sum119896isin119870

119888minus1

120596minus120574119896

[(1199112minus 119911

1+ radicΔ

119896

2)

120572

(1199111minus 119911

2+ radicΔ

119896

2)

120573

sdot 119891 (1199111+ 119911

2+ radicΔ

119896

2)


sdot (1199112minus 119911

1minus radicΔ

119896

2)

120572


2minus radicΔ

119896

2)

120573

119891(1199111+ 119911

2minus radicΔ

119896

2)]

=

infin



eminus119894120587119888(119899+1)[120579 (119911)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911minus1199112


(120579 (119911)

(119911minus1199112) (119911minus119911

1))


1205791015840

(119911)119891 (119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199111

(55)

where

Δ119896= (119911

1minus 119911

2)2

+ 4119881 (120596119896

120579 (119911))

119881 (119911) =

infin

sum119903=1

119863119903minus1

119911[119902 (119911)]

minus119903

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

119911119903

119903

120579 (119911) = (119911 minus 1199111) (119911 minus 119911

2) 119902 ((119911 minus 119911

1) (119911 minus 119911

2))

(56)


(119911 minus 1199111)(119911 minus 119911

2)) and 119911


119891 (119911) = 119888119911minus120573

(119911 minus 1199111)minus120572

times

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot119863minus120572+119888119899+120574

119911119911


(119911 + 119908 minus 1199111)119891(119911)

10038161003816100381610038161003816119911=1199111

(119908=119911)

(57)



119863120572

119911119911

119901+119902

119906 (119911) V (119911)

=

infin


(120572

120574 + 119899)119863

120572minus120574minus119899

119911119911

119901

119906 (119911)119863120574+119899

119911119911

119902V (119911) (58)




119863120572

119911119911

120572+120573minus1

119906 (119911) V (119911)


sdot

infin


119863120572+]+1minus119899119911



(60)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 (1120582))]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot 119865(119896minus1)



where 120582 gt 0 and 119865(119899)119863


119865(119899)

119863[119886 119887

1 119887

119899 119888 119909

1 119909

119899]

=

infin

sum1198981119898119899=0

(119886)1198981+sdotsdotsdot+119898

119899

(1198871)1198981

sdot sdot sdot (119887119899)119898119899

(119888)1198981+sdotsdotsdot+119898

119899

1199091198981

1


119909119898119899

119899

119898119899

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091198991003816100381610038161003816 lt 1)

(62)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119896

(119911119896minus1)120572

sdot119911119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) (119911119896minus1

)120572

sdot

119896minus1

prod119904=1

(1 minus119911

119908eminus2120587119894119904119896)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(63)




119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

(64)



119896

(119911119896minus1)120572

119911

119874120572

120573

times

1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119911119899+(119896minus1)120572

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


sdot((119911119908) eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119908=119911

=119896

120582Γ (119904) (119911119896minus1)120572

times

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981



sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

times119911

119874120572

120573119911


119896minus1+119899+(119896minus1)120572

=119896

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


times(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

sdotΓ (120573) Γ (119898


119896minus1+ 119899 + 119896120572)

Γ (120572) Γ (120573 + 1198981+ sdot sdot sdot + 119898

119896minus1+ 119899 + 119896120572)


=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 1120582)]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(119899+119896120572)1198981+sdotsdotsdot+119898

119896minus1

(1+120572minus120573)1198981



119896minus1

sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

(65)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


119911minus1198881198990(119911 minus 119911

0)119888119899

Γ (119888119899 + 1) Γ (1 minus 119888119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(1003816100381610038161003816119911 minus 1199110

1003816100381610038161003816 =100381610038161003816100381611991101003816100381610038161003816 120582 gt 0)

(66)


Proof Setting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


0 we

have

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


(119911 minus 1199110)119888119899

Γ (1 + 119888119899)

sdot119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

(67)


0| = |119911

0|


119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

=119911minus1198881198990

Γ (1 minus 119888119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(68)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(69)


119911 =1199111

2+1199111



Proof By taking 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 9 with 119911

2= 0 120583 = 120572 ] = 120573 and 0 lt 119888 ≦ 1 we find

that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888(119911 minus 1199111)minus120573

119911minus120572

1199111

sdot

infin



sdot (119911 minus 119911

1

119911)119888119899+120574

sdot119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

(71)



119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120572+120573minus1

1

Γ (120572 + 119888119899 + 120574)

Γ (120572 + 120573)

times Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582)

(72)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573+120574

(119911 minus 1199111)minus120572+120574

119911120573+120572minus2120574minus1

1

sdot

infin


sin [(120573 minus 119888119899 minus 120574) 120587] e119894120587119888(119899+1)

sin [(120573 + 119888 minus 120574) 120587] Γ (1 minus 120572 + 119888119899 + 120574)

sdot (119911(119911 minus 119911

1)

11991121

)

119888119899

Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdot [(119911 minus 1199111)Φ

(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (120573 minus 119888119899 minus 120574

120572 + 120573 minus 2119888119899 minus 2120574) 119911

1

sdotΦ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)]

(73)


119911 =1199111

2+1199111



Proof Putting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 10 with 119911

2= 0 0 lt 119888 ≦ 1 119902(119911) = 1 and 120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2) we find that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573

(119911 minus 1199111)minus120572

sdot

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

(75)


119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

= 119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

+ (119911 minus 1199111)119863

minus120572+119888119899+120574

119911


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120573+120572minus2119888119899minus2120574

1


Γ (1 + 120573 + 120572 minus 2119888119899 minus 2120574)

sdot Φ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (119911 minus 119911

1

1199111

)Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdotΦ(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582) )

(76)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

=Γ (120591) Γ (1 + ] minus 120591) sin (120574120587)

120587

sdot

infin


(minus1)119899

Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)


(77)



12058211205821199011205831120583119902


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=

infin


(] minus 120591120574 + 119899


119911]minus1

sdot 119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(78)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)



119911]minus1 =

Γ (])Γ (120591 + 120574 + 119899)

119911120591+120574+119899minus1

119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911minus120574minus119899

Γ (1 minus 120574 minus 119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(79)



Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

= (Γ (120591) Γ (1 + ] minus 120591) sin (120573120587)




sdotsin (120579120587)

sin [(120579 + 120574) 120587]

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(80)



120572 = ] minus 120591 119906 (119911) = 119911120591minus120573

V (119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(81)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


sdot

infin


119863]minus120591+120574+1minus119899119911




119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(82)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

119863]minus120591+120574+1minus119899119911


=Γ (] minus 120579 minus 119899)


119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (120579 + 119899)

Γ (1 + 120579 + 120574)

times 119911120579+120574

Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(83)





lim119887rarr0

11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)] = 120582Γ (119904)

(120582 gt 0)

(84)



Φ(120588112058811990113120590

112059011990213)

12058211205821199011205721205831120583119902120573

(119911 119904 119886)

=3Γ (120573) Γ (3120572)

Γ (120572) Γ (120573 + 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

(3120572)119899

(120573 + 120572)119899

119911119899

(119899 + 119886)119904


(85)


1198651[119886 119887

1 119887

2 119888 119909

1 119909

2]

=

infin

sum11989811198982=0

(119886)1198981+1198982

(1198871)1198981

(1198872)1198982

(119888)1198981+1198982

1199091198981

1

1198981

1199091198982

2

1198982

(max 100381610038161003816100381611990911003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 lt 1)

(86)




1= 120583 in



Θ120582

120583(119911 119904 119886 119887)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(111)

120583120572+119888119899+120574120572+120573(119911

1 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(87)


119911 =1199111

2+1199111





Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886)

=Γ (120591) Γ (1 + ] minus 120591)



times (Γ (1 + 120574 + 120579) sin [(] minus 120591 + 120573) 120587]


sdot

infin


Γ (]minus120579minus119899) Γ (120579+119899)Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120582

(119911 119904 119886)





References













































Algebra




Geometry












Operations Research








Contents





















Toka Diagana










1 Introduction




119886(119899 119896) and 119878



(119905)119899=

119899

sum119896=0

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

(119905 minus 119886)119896

(1)

(119905 minus 119886)119899

=

119899

sum119896=0

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(119905)119896 (2)


119904119886(119899 119896) =

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

119878119886(119899 119896) =

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(3)


119904119886(119899 + 1 119896) = 119904

119886(119899 119896 minus 1) + (119886 minus 119899) 119904

119886(119899 119896) (4)

119878119886(119899 + 1 119896) = 119878

119886(119899 119896 minus 1) + (119896 minus 119886) 119878

119886(119899 119896) (5)


119904119886(0 0) = 1 119904

119886(119899 0) = (119886)

119899 119904

119886(0 119896) = 0 119899 119896 = 0

119878119886(0 0) = 1 119878

119886(119899 0) = (minus119886)

119899

119878119886(0 119896) = 0 119899 119896 = 0

(6)


1+ 119883


119896of





119875 [119882 = 119899] =119896

(1 minus 120579)minus119904

(minus log (1 minus 120579))119896

120579119899

119899(minus1)

119899minus119896

119904minus119904(119899 119896)

119875 [119885 = 119899] =119896

119890119898 (119890119897 minus 1)119896

119897119899

119899(minus1)

119899minus119896

119878minus119898119897

(119899 119896)

(7)






119870(1)

119899119903= 119903 + [log(119899 minus 1


1

119903+ 119900 (1)] (8)


2 Explicit Formula



119865119899(119909) = prod

119899


1198651015840

119899(119909) = sum


119899minus1le119899

119894isinN11989911989511198952119895119899minus1

1198911015840

119894(119909)

119899minus1

prod119896=1

119891119895119896(119909) (9)


1 2

119889

119889120585(120585119895 + 119886)

1= 119895

119889

119889120585(120585119895 + 119886)

2= (120585119895 + 119886) 119895 + (120585119895 + 119886 minus 1) 119895

(10)


1

119895119899119889

119889120585(120585119895 + 119886)



119899minus1

prod119896=1

(120585 +119886 minus 119895

119896

119895) (11)



1

119895119899119889119896

119889120585119896(120585119895 + 119886)


119899minus119896le119899minus1

119896

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(12)


1

119895119899119889119898

119889120585119898(120585119895 + 119886)


119899minus119898le119899minus1

119898

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(13)

Then

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899


119899minus119898le119899minus1

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(14)




119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895) = sum



prod119902=1

(120585 +119886 minus 119894

119902

119895)

(15)

119894119902

isin 1198951 1198952 119895

119899minus119898 has ( 119899minus119898





119902=1(120585 + (119886 minus





119895119899minus119898minus1





119902=1(120585 + (119886 minus 119895


of (1119895119899)(119889119898+1119889120585119898+1)(120585119895 + 119886)119899 Thus we have

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899



119898 + 1

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)



119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(16)

Lemma 2 Let 119904(119899 119896 119886) =

(1119896)lim120585rarr0

((sum119896

119895=0(minus1)

119896minus119895

( 119896119895) (119889119896119889120585119896)(120585119895 + 119886)

119899)119896)

Then


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (17)

Proof Using Lemma 1

119904 (119899 119896 119886)

=1

119896lim120585rarr0

((

119896

sum119895=0


(119896

119895) 119895119899


119899minus119896le119899minus1

119896

times

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)) (119896)

minus1

)

(18)



(120585 + (119886 minus 119895119902)119895) = (1119895119899minus119896)prod

119899minus119896

119902=1(120585119895 + 119886 minus 119895

119902)


1

119896lim120585rarr0

[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(120585119895 + 119886 minus 119895119902)]

]

=1

119896[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)]

]

(19)

which boils down to


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (20)

since

1

119896

119896

sum119895=0


(119896

119895) 119895119896

= 119878 (119896 119896) = 1 (21)



119904119886(119899 119896) = 119904 (119899 119896 119886) = sum


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (22)

Proof We know that


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (23)



119899+1minus119896)







119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (24)

is equal to


119899minus119896+1le119899minus1

119899minus119896+1

prod119902=1

(119886 minus 119895119902) + (119886 minus 119899)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)

(25)

This implies that

119904 (119899 + 1 119896 119886) = 119904 (119899 119896 minus 1 119886) + (119886 minus 119899) 119904 (119899 119896 119886) (26)



119886(119899 119896)


120582 = (1205821ge 120582


119896) (27)


119894


14= 11989115

= 11989123

= 11989131

= 11989142

= 1 119891119894119895= 0 elsewhere

(1 le 119895 le 120582119894)

0 0 0 1 10 0 11 0 00 10

(28)



(119886119894)119894ge0



1198860 1198861 119886

ℎ then |119879119889119860(ℎ 119903)| = ( ℎ+1




120596-tableau is











119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)


120596 (|119888|) (29)




is equal to 119904119886(119899 119896)



0 1 2 119899 minus 1 and

120596119860(120601) = prod

119888isin120601

120596 (|119888|)

=

119899minus119896

prod119894=1

(119886 minus 119895119894) 119895

119894isin 0 1 2 119899 minus 1

(30)

If 119886 = 1198861+ 1198862for some 119886

1and 119886

2 then with 120596lowast(119895) = 119886

2minus 119895

120596119860(120601) =

119899minus119896

prod119894=1

(1198861+ 120596lowast

(119895119894))

=

119899minus119896

sum119903=0

119886119899minus119896minus119903

1sum


119903le119895119899minus119896

119903

prod119894=1

120596lowast

(119902119894)

(31)






may write

120596119860(120601) = sum

120595isin119861120601

120596119860(120595) (32)


120596119860(120595) = 119886

119899minus119896minus119903

1

119903

prod119894=1

120596lowast

(119902119894) (33)

where 1199021 1199022 119902

119903isin 119895

1 1198952 119895


written as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

sum120595isin119861120601

120596119860(120595) (34)



1 1198952 119895

119899minus119896



(119899

119896)(

119899 minus 119896

119903) (35)





( 119899119896) ( 119899minus119896

119903)

( 119899119903)

= (119899 minus 119903

119896) (36)


119904119886(119899 119896) =

119899minus119896

sum119903=0

(119899 minus 119903

119896) 119886119899minus119896minus119903

1sum120595isin119861119903


120596lowast

(|119888|) (37)



119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

1sum



120596lowast

(|119888|) (38)

Clearly 119861119899minus119895



tity

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

11199041198862

(119899 119895) (39)


1and 119886

2





119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (40)






2isin 1198791198891198602(119895 minus 1 119895 minus 119899+ 119896)


2 we obtain an


119860

(119898 +


sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601)

=

119899

sum119896=0

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011)

times

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

(41)


Note that

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)


119895minus119899+119896le119898+119895minus1

119895minus119899+119896

prod119902=1

(119886 minus 119892119902)



119895minus119899+119896

prod119902=1

(119886 minus (119898 + 119892119902))

= 119904119886minus119898

(119895 119899 minus 119896)

(42)


sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011) = 119904

119886(119898 119896)

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601) = 119904

119886(119898 + 119895 119899)

(43)

Thus

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (44)



convolution formula

119904119886(119899 + 1119898 + 119895 + 1) =

119899

sum119896=0

119904119886(119896 119899) 119904

119886minus(119896+1)(119899 minus 119896 119895) (45)

Proof Let


in 1198601= 0 1 119896 minus 1


are in 1198602= 119896 + 1 119899








21199092 + sdot sdot sdot + 119886

119899119909119899 has only

real zeros then

1198862

119896ge 119886119896+1

119886119896minus1

119896

119896 minus 1

119899 minus 119896 + 1

119899 minus 119896(119896 = 2 119899 minus 1) (46)


sum119896=0

119904119886(119899 119896) (119905 + 119886)

119896

(47)



119896=0119904119886(119899 119896)119905119896 are 119886 119886minus1 119886minus



119899


and hence unimodal


⟨119905⟩119899=

119899

sum119896=0


119904119886(119899 119896) (119905 + 119886)

119896

(48)



119904119886(119899 119896) = (minus1)

119899minus119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902) (49)


119886(119899 119896)| as

1003816100381610038161003816119904119886 (119899 119896)1003816100381610038161003816 = (minus1)

119899minus119896

119904119886(119899 119896)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902)

(50)



lt 1199062


infin

sum119894=1

1

119906119894

= infin

infin

sum119894=1

1

1199062119894

lt infin (51)




which sum119899119896


119870119899= 119899 minus [

119899

sum119894=1

1

119906119894

minus

119899

sum119894=1

1

1199062119894

(1 +1

119906119894

)

minus1

+ 119900 (1)] (52)


1

1+1

2+ sdot sdot sdot +

1

119899= log 119899 + 120574 + 119900 (1) (53)



119886(119899 119896)|

119899

119896=0



119899


mation

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (54)




Denoting by 119896119886119899



1= 119887 + 0 119906

2= 119887 + 1 119906


119896119886119899

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894)2(1 +

1

119887 + 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894) (119887 + 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894 + 1+ 119900 (1)]

(55)


119899minus1

sum119894=0

1

119887 + 119894 + 1= log (119887 + 119899) minus log 119887 (56)

From this we get

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (57)



119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (58)



1)(119886 minus


119899minus119896)where 119895


1 By Theorem 10 119904119886(119899 119896) = sum

119899119899minus119896 Thus with 119906

1= 119886 119906

2=

119886 minus 1 119906119899minus1

= 119886 minus 119899 + 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894)2(1 +

1

119886 minus 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894) (119886 minus 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119886 minus 119894 + 1+ 119900 (1)]

(59)


119899119896 0 1 2 30 11 12 2 13 6 6 14 24 35 10 15 120 225 85 156 720 1624 735 1757 5040 13132 6769 19608 40320 109584 118124 672849 362880 1026576 1172700 72368010 3628800 10628640 12753576 8409500


119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (60)


9


119896minus19

= [log(1 + 8

1) + 119900 (1)]

= [log 9 + 119900 (1)]

asymp 2

(61)


10


1198961010

= [log( 10 + 1

10 minus 10 + 1) + 119900 (1)]

= [log 11 + 119900 (1)]

asymp 2

(62)







(63)


Table 2 Values of 11990410(119899 119896)

119899119896 0 1 2 30 11 10 12 90 19 13 720 242 27 14 5040 2414 431 345 30240 19524 5000 6356 151200 127860 44524 81757 604800 662640 305956 772248 1814400 2592720 1580508 5376289 3628800 6999840 5753736 265576410 3628800 10628640 12753576 8409500


119904(119899 119896) 0 1 2 3 4 50 11 0 12 0 1 13 0 2 3 14 0 6 11 6 15 0 24 50 35 10 16 0 120 274 225 85 157 0 720 1764 1624 735 1758 0 540 13068 13132 6769 19609 0 40320 109584 118124 67284 2244910 0 362880 1026576 1172700 723680 269325





2= 2 119906


119896119899= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

1

1198942(1 +

1

119894)minus1

+ 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

(1

119894minus

1

119894 + 1) + 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894 + 1+ 119900 (1)]

(64)


119899minus1

sum119894=1

1

1 + 119894= log 119899 minus log 1 + 120574 + 119900 (1) (65)

Therefore we have

119896119899= [log 119899 + 120574 + 119900 (1)] (66)



7

119896=0occurs at

1198967= [log 7 + 120574 + 119900 (1)]

= [195 + 05772 + 119900 (1)]

= [253 + 119900 (1)]

asymp 2

(67)


11989610

= [log 10 + 120574 + 119900 (1)]

= [230 + 05772 + 119900 (1)]

= [28772 + 119900 (1)]

asymp 3

(68)




Acknowledgment


References
















1 Introduction















2 New Instructions







x

f(x)

i

(ii) and (iii)1

c













2119888 and



















2minus119890


minus119873

2minus119873

] (1)









2119909 multiplica-








1sdot finite

1+ finite

2= 2 (2)








2120583 in the case






2((maximum



Inputsign exp mant

Drop leading zeros

numZeros + 1


Add

Switch

s e12 m51

lookup output


exp

Add 1

expP1

adjustedE

Switch

bitRot


Add 0x7ff

Add 0x3ff

Switch

Concatenate bit

xor

Add drop carry

moreThan22Zeros

isNot12

Concatenate bit

51

42

42 + 1

44

30

31

52

52

52

52


Add



(c)

(b)

(d)

(e)

(f)

(g)


(h)

(i)

(j)

(k)

leading12Mant

(a)

(p)

(r)

(o)

(l)

(m)

(n)

(q)

(s)

(t)

(u)

(v)

5

sim

sim

0x7ffSubs

Subs 0b0 0




12-bit lookup

12-bit lookup

ge1

e(last 10bits)lt1

sim


0b10 0Subs

Subs 0b0 0

low 10bits4

6

11

11

11

11

12

Lower 10






2119872

recip 3 113 376 0500 2048 3 infin






3 Evaluation





4 Conclusion




(a)


recip (2minus119890

119888)ext (2minus119890



119888)sat

sqrt (2minus119890

119888)ext 21198902

119888 0infin 0 NaN 0 0 0 NaN




(b)


minusinfin 119888 119888 0 0

infin 119888 0 119888 0

NaN 119888 119888 119888 0

(c)



minusinfin119891

minusinfin119891

infin infin119891

infin119891

infin119891

infin119891

NaN0

NaN119891

NaN119891

NaN119891

NaN119891

NaN1

2119891

2119891

2119891

2119891

NaN2

1radic2119891

1radic2119891

1radic2119891

1radic2119891

NaN3

0119891

0119891

0119891

0119891

recipdiv

sqrtrsqrtcbrt

rcbrtqdrt

rqdrtlog2

loglog1pexp2

expexpm1

atan2atanasinacos

sincostan

0 5 10 15 20 25 30







Acknowledgments


References

























1 Introduction








[119865 (119905)]119896

= sum119899gt0

119865 (119899 119896) 119905119899

(1)





119860Δ

(119899 119896) = 120572119896

119865Δ

(119899 119896) (2)



119860Δ

(119899 119896) = 120572119899

119865Δ

(119899 119896) (3)


119891(119899)119905119899 119877(119905) = sum119899gt0

119903(119899)119905119899


119860Δ

(119899 119896) =

119899

sum119898=119896

119865Δ

(119899119898)119866Δ

(119898 119896) (4)


119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899



119886(119899)119905119899

are

119886 (119899) =

119899

sum119896=1

119865Δ

(119899 119896) 119903 (119896)

119886 (0) = 119903 (0)

(5)




119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899


119899ge0119886(119899)119905119899 are

119886 (119899) =

119899

sum119896=0

119891 (119896) 119903 (119899 minus 119896) (6)




119910119899(119909) =

119899

sum119896=0

(119899 + 119896)

(119899 minus 119896)119896(119909

2)119896

(7)


119901119899(119909) = 119909

119899

119910119899minus1

(1

119909) (8)


119901119899(119909) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)(1

2)119899minus119896

119909119896 (9)


sum119899=0

119901119899(119909)

119899119905119899


(10)




119862Δ

(119899 119896) =119896

119899(2119899 minus 119896 minus 1

119899 minus 1) (11)



2)

119861 (119905) = 119890119905

=

infin

sum119899=0

1

119899119905119899

(12)


119860Δ

(119899 119896) = (2119909)119896

(1

2)119899

119862Δ

(119899 119896) = 2119896minus119899

119909119896119896

119899(2119899 minus 119896 minus 1

119899 minus 1)

(13)


119899=0119887(119899)119905119899

are equal to

119887 (119899) =1

119899 (14)


119901119899(119909) = 119899

119899

sum119896=1

119860Δ

(119899 119896) 119887 (119896) =

119899

sum119896=1

(2119899 minus 119896 minus 1)


119909119896

(15)




119898119899+1

(119909 120573 119888) =(119888 minus 1)

119899

119888119899

sdot ((119909 minus 119887119899)119898119899(119909 120573 119888) minus 120582

119899119898119899minus1

(119909 120573 119888))

(16)

where

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) = 119909 minus 119887

0

119887119899=

(1 + 119888) 119899 + 120573119888

1 minus 119888

120582119899=

119888119899 (119899 + 120573 minus 1)

(1 minus 119888)2

(17)


1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) =

119909 (119888 minus 1) + 120573119888

119888

1198982(119909 120573 119888)

=1199092 (119888 minus 1)

2

+ 119909 ((2120573 + 1) 1198882 minus 2120573119888 minus 1) + (120573 + 1) 1205731198882

1198882

(18)


infin

sum119899=0

119898119899(119909 120573 119888)

119899119905119899

= (1 minus119905

119888)119909


(19)





119860 (119905) = (1 minus119905

119888)119909

=

infin

sum119899=0

(119909

119899) (minus1)

119899

119888minus119899

119905119899


=

infin

sum119899=0


119899) (minus1)

119899

119905119899

(20)


suminfin

119899=0119886(119899)119905119899 and 119861(119905) = sum

infin


follows

119886 (119899) = (119909

119899) (minus1)

119899

119888minus119899

119887 (119899) = (minus119909 minus 120573

119899) (minus1)

119899

(21)


119898119899(119909 120573 119888) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896)

= (minus1)119899

119899

119899

sum119896=0

(119909

119896)(


119899 minus 119896) 119888minus119896

(22)



119872119899+1

(119909 120575 120578) = (119909 minus 119887119899)119872119899(119909 120575 120578) minus 120582

119899119872119899minus1

(119909 120575 120578)

(23)

where1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 119887

0

119887119899= (2119899 + 120578) 120575

120582119899= (1205752

+ 1) 119899 (119899 + 120578 minus 1)

(24)


1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 120575120578

1198722(119909 120575 120578) = 119909

2

minus 1199092120575 (1 + 120578) + 120578 ((120578 + 1) 1205752

minus 1)

(25)


infin

sum119899=0

119872119899(119909 120575 120578)

119899119905119899

= ((1 + 120575119905)2

+ 1199052

)minus1205782


1 + 120575119905))

(26)




119860 (119905) = ((1 + 120575119905)2

+ 1199052

)minus1205782

= (1 + 2120575119905 + (1205752

+ 1) 1199052

)minus1205782

119861 (119905) = exp (119909tanminus1 ( 119905

1 + 120575119905))

(27)


1(1198602(119905)) and we expand 119860


theorem

1198601(119905) = (1 + 119905)

minus1205782

=

infin

sum119899=0

(minus120578

2119899

) 119905119899

1198602(119905) = 2120575119905 + (120575

2

+ 1) 1199052

(28)


infin

119899=01198861(119899)119905119899

are

1198861(119899) = (

minus120578

2119899

) (29)


1198862119896minus119899

119887119899minus119896

(119896

119899 minus 119896) (30)


119860Δ

2(119899 119896) = (2120575)

2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896) (31)


infin

119899=0119886(119899)119905119899

119886 (0) = 1198861(0)

119886 (119899) =

119899

sum119896=1

119860Δ

2(119899 119896) 119886

1(119896)

(32)


119886 (0) = 1

119886 (119899) =

119899

sum119896=1

(2120575)2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896)(

minus120578

2119896

) (33)


1(1198612(119905)) where

1198611(119905) = 119890

119905

=

infin

sum119899=0

1

119899119905119899

1198612(119905) = 119909tanminus1 ( 119905

1 + 120575119905)

(34)



functions 1199091198621(1198622(119905)) where

1198621(119905) = tanminus1 (119905)

1198622(119905) =

119905

1 + 120575119905

(35)


((minus1)(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896] (36)

where [119895



119862Δ

1(119899 119896) = ((minus1)

(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

sdot

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896]

(37)


(119899 minus 1

119896 minus 1) 119886119896

119887119899minus119896

(38)


to

119862Δ

2(119899 119896) = (

119899 minus 1

119896 minus 1) (minus120575)

119899minus119896

(39)


2(119905)

119861Δ

2(119899 119896) = 119909

119896

119899

sum119898=119896

119862Δ

2(119899119898) 119862

Δ

1(119898 119896) (40)


infin

119899=01198871(119899)119905119899

are defined by

1198871(119899) =

1

119899 (41)


infin

119899=0119887(119899)119905119899

119887 (0) = 1198871(0)

119887 (119899) =

119899

sum119896=1

119861Δ

2(119899 119896) 119887

1(119896)

(42)


119887 (0) = 1

119887 (119899) =

119899

sum119896=1

119909119896

119899

119899

sum119898=119896

(119899

119898)120575119899minus119898

(1 + (minus1)

119898+119896

(minus1)(119898+119896)2minus119899

)

sdot

119898

sum119895=119896


(119895 minus 1)(119898

119895)[

119895

119896]

(43)


119872119899(119909 120575 120578) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896) (44)



Acknowledgment


References





































119892 (119911) =1

119911+ 1198870+

infin

sum119899=1

119887119899119911119899 (1)




1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911) (2)






119899| le

(119861 minus 119860)(119899 + 1) for 119892 isin Σ[119860 119861]



ℎ (119908) = 119892minus1

(119908) =1

119908+

infin

sum119899=0

119861119899119908119899

=1


1

119899119870119899

119899+1(1198870 1198871 119887

119899) 119908119899

(3)

where 119908 isin D

119870119899

119899+1(1198870 1198871 119887

119899)

= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)


3119887119899minus4

0(1198874+ 311988711198872)

+ sum119895ge5

119887119899minus119895

0119881119895

(4)


variables 1198871 1198872 119887

119899





119899|



119899| and




119899=1119901119899119911119899 be so that 119877119890(119901(119911)) gt 0


100381610038161003816100381610038161199012+ 1205721199012

1

10038161003816100381610038161003816le 2 + 120572

1003816100381610038161003816119901110038161003816100381610038162

(5)



119899=1119888119899119911119899


100381610038161003816100381610038161198882+ 1205741198882

1

10038161003816100381610038161003816le 1 + (120574 minus 1)

1003816100381610038161003816119888110038161003816100381610038162

(6)

Proof Write

119901 (119911) =[1 + 120593 (119911)]

[1 minus 120593 (119911)] (7)

where 119901(119911) = 1+suminfin



1= 21198881and 119901

2=

2(1198882+ 11988821)


2= 2(1198882+ 11988821) in (5) we

obtain100381610038161003816100381610038162 (1198882+ 1198882

1) + 120572 (2119888

1)210038161003816100381610038161003816le 2 + 120572

10038161003816100381610038162119888110038161003816100381610038162 (8)

or100381610038161003816100381610038161198882+ (1 + 2120572) 119888

2

1

10038161003816100381610038161003816le 1 + 2120572

1003816100381610038161003816119888110038161003816100381610038162

(9)




(119894)10038161003816100381610038161198870

1003816100381610038161003816 le

119861 minus 119860


119861 minus 119860 119900119905ℎ119890119903119908119894119904119890

(119894119894)10038161003816100381610038161198871

1003816100381610038161003816

le

119861 minus 119860


2 (119861 minus 119860)|1198870|2

119894119891 0 le 119860 le 1 minus 2119861

119861 minus 119860

2 119900119905ℎ119890119903119908119894119904119890

(119894119894119894)100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2

(119894V) 10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 119894119891 119887

119896= 0 119891119900119903 0 le 119896 le 119899 minus 1

(10)


1199111198921015840 (119911)

119892 (119911)= minus1 minus

infin

sum119899=0

119865119899+1

(1198870 1198871 1198872 119887

119899) 119911119899+1

(11)

where 119865119899+1

(1198870 1198871 1198872 119887




5=



119865119899+1

(1198870 1198871 119887

119899)


119899+1=119899+1

119860 (1198941 1198942 119894

119899+1) 1198871198941

01198871198942

1sdot sdot sdot 119887119894119899+1

119899

(12)

where

119860 (1198941 1198942 119894

119899+1)

= (minus1)(119899+1)+2119894

1+sdotsdotsdot+(119899+2)119894

119899+1

(1198941+ 1198942+ sdot sdot sdot + 119894

119899+1minus 1) (119899 + 1)

(1198941) (1198942) sdot sdot sdot (119894

119899+1)

(13)



119908ℎ1015840 (119908)


infin

sum119899=1

119865119899+1

(1198610 1198611 1198612 119861

119899) 119908119899+1

(14)

where119865119899+1

(1198610 1198611 1198612 119861


119899 + 1 given by

119865119899+1

= minus119899 (119899 minus (119899 + 1))

119899 (119899 minus 2119899)119861119899

0

minus119899 (119899 minus (119899 + 1))


01198611

minus119899 (119899 minus (119899 + 1))


01198612

minus119899 (119899 minus (119899 + 1))


0

sdot (1198613+119899 minus (2119899 minus 3)

21198612


119861119899minus119895

0119870119895

(15)


variables 1198611 1198612 119861

119899minus1and

1198610= minus1198870 119861

119899= minus

1

119899119870119899

119899+1(1198870 1198871 119887

119899) (16)


1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198881 1198882 119888

119899 119861) 119911119899

(17)

119908ℎ1015840 (119908)

ℎ (119908)= minus

1 + 119860120595 (119908)

1 + 119861120595 (119908)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198891 1198892 119889

119899 119861) 119908

119899

(18)


119899=1119888119899119911119899 and 120595(119908) = sum

infin

119899=1119889119899119908119899 are two



119899(1198961 1198962 119896

119899 119861) are given by

119870119901

119899(1198961 1198962 119896

119899 119861)

=119901

(119901 minus 119899)119899119896119899

1119861119899minus1

+119901


11198962119861119899minus2

+119901


11198963119861119899minus3

+119901


1

sdot [1198964119861119899minus4

+119901 minus 119899 + 3

21198962

2119861]

+119901


1

sdot [1198965119861119899minus5

+ (119901 minus 119899 + 4) 11989621198963119861]

+ sum119895ge6

119896119899minus119895

1119883119895

(19)


variables 1198962 1198963 119896

119899


minus119865119899+1

(1198870 1198871 1198872 119887

119899) = (119860 minus 119861)119870

minus1

119899+1(1198881 1198882 119888

119899+1 119861)

(20)


minus 119865119899+1

(1198610 1198611 1198612 119861

119899)

= (119860 minus 119861)119870minus1

119899+1(1198891 1198892 119889

119899+1 119861)

(21)


1198870= (119860 minus 119861) 119888

1

minus1198870= (119860 minus 119861) 119889

1

(22)

21198871minus 1198872

0= (119861 minus 119860) (119888

2

1119861 minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (119889

2

1119861 minus 1198892)

(23)




0and then adding

them gives

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus 1198611198882

1minus 1198611198892

1) (24)



21198872

0= (119861 minus 119860) (119888

2+ 1198892minus

2119861

(119861 minus 119860)21198872

0) (25)


sides we obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(119861 minus 119860)

2

(1198882+ 1198892)

2 (2119861 minus 119860)le

119861 minus 119860

radic2119861 minus 119860 (26)



119861 minus 119860



we obtain

41198871= (119861 minus 119860) [(119889

2minus 1198882) minus (119889

2

1minus 1198882

1) 119861] = (119861 minus 119860) (119889

2minus 1198882)

(28)


100381610038161003816100381611988711003816100381610038161003816 le

119861 minus 119860

2 (29)


21198871= (119861 minus 119860) (119889

2+ 1198601198892

1minus 2119861119889

2

1) (30)


100381610038161003816100381611988711003816100381610038161003816 le

1

2(119861 minus 119860) (

100381610038161003816100381610038161198892+ 1198601198892

1

10038161003816100381610038161003816+ 2119861

1003816100381610038161003816119889110038161003816100381610038162

)

le1

2(119861 minus 119860) (1 + (119860 minus 1)

1003816100381610038161003816119889110038161003816100381610038162

+ 211986110038161003816100381610038161198891

10038161003816100381610038162

)

=119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

(31)




2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

lt119861 minus 119860

2 (32)


21198871minus 1198872

0= (119861 minus 119860) (

119861

(119861 minus 119860)21198872

0minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (

119861

(119860 minus 119861)21198872

0minus 1198892)

(33)


100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2 (34)


(119899 + 1) 119887119899= (119860 minus 119861) 119888

119899+1 (35)


obtain

10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 (36)







0|2 given by






References



























1 Introductions


119891 (119911) = 119911 +

infin

sum120581=2

119886120581119911120581

(1)


119867[119886 119899] = 119901 isin 119867 119901 (119911) = 119886 + 119886119899119911119899

+119886119899+1

119911119899+1


(2)


119865 (119911) = 119911120573

+

infin

sum120581=2

120582120581119886120581119911120573+120581minus1

120573 isin N

120582120581= 120573120581minus1

119911 =1

120573 120573 |119911| lt 1

(3)


(119891 lowast 119892) (119911) =

infin

sum119896=0

119886119896119887119896119911119896

(4)


119896=0119886119896119911119896 and 119892(119911) = sum

infin

119896=0119887119896119911119896 are analytic in

119880

Let 120595(119911) = 119911120573 + sum

infin



120595 (119911) lowast119911120573

1 minus 120573119911 119911 =

1

120573 120573 isin N 120573 |119911| lt 1 (5)



119876(120485) (120572 120591 120574 120573)

= 119865 (119911) isin 119867 (120573) R119890

times120572[119865(119911)]

(120485)

[119911120573](120485)

+120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

gt 120574 119911 isin 119880

(6)

where

119911 = 0 120573 gt 120485 120485 isin N cup 0 also 120572 + 120591 = 120574 (7)







119891 (119911) =1

120572 + 120591int|119909|


times

infin

sum120581=0

(120572 + 120591) 119909120581

119911120581+1

(120581 + 1) 120591 + 120572] 119889120583 (119909)

(8)




Θ120572120573

(119911) = (120573int119911

0

119905120573minus120572minus1

[119891 (119905)]120572

119892 (119905) 119889119905)

1120573

(10)






119901 (119911) = int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909) (|119911| lt 1) (11)



119892 (119911) +1199111198921015840 (119911)

119888≺ ℎ (119911) (12)

then

119892 (119911) ≺ 119902 (119911) =119888119911minus119888120581

120581int119911

0

119905(119888minus120581)120581

ℎ (119905) 119889119905 ≺ ℎ (119911) (13)



1199111199011015840 (119911)

119901 (119911)≺ ℎ (119911) (14)

then

119901 (119911) ≺ 119902 (119911) = 119886 exp [1

119899int119911

0

ℎ (120585) 120585minus1

119889120585] (15)




(ii) |1199011015840(119911)| le 2R119890119901(119911)(1 minus 1199032)


100381610038161003816100381610038161003816100381610038161003816

1199111199011015840 (119911)

119901 (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 119901 (119911)

(1 minus 119903)2

(16)




(120485)

(120572120591120574120573)




119911120485

[119865 (119911)](120485)

=

120573

119875120485

(120572 + 120591)int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120573 minus 120485) (120572 + 120591) 119909120581119911120573+120581

(120572 + 120591) (120573 minus 120485) + 120581120591] 119889120583 (119909)

(17)

where 119899119875119903




119901 (119911)

=

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

isin 119867 [1 119899]

(18)

ByTheorem 3

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

= int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909)

(19)


[119865 (119911)](120485+1)

[119911120573](120485+1)

+120572[119865 (119911)]

(120485)

120591[119911120573](120485)

=1

120591int|119909|=1

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909119911

1 minus 119909119911119889120583 (119909)

(20)

which yields

119911((120572+120591)(120485minus120573)+120591)120591

int119911

0

[[119865 (120585)]

(120485+1)

[120585120573](120485+1)

+120572[119865 (120585)]

(120485)

120591[120585120573](120485)

]


119889120585

=1

120591int|119909|=1

119911((120572+120591)(120485minus120573)+120591)120591

times int119911

0

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909120585

1 minus 119909120585


119889120585 119889120583 (119909)

(21)



shows that 119865(119911) isin 119876(120485)(120572 120591 120574 120573)



119865 (119911) =1

120572 + 120591int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581119911120573+120581

120572120573 + 120591 (120581 + 120573)] 119889120583 (119909)

(22)


1




10038161003816100381610038161198861205811003816100381610038161003816 le 119867 (120572 120591 120574 120573) (23)

where

119867(120572 120591 120574 120573) =2 (120572 + 120591 minus 120574) (120573 minus 120485) [

120573

119875120485]

120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(24)


119911120485

[119865119909(119911)](120485)

120573119875120485

= 119911120573

+

infin

sum120581=2


(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)(|119909| = 1)

(25)




119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [[120581 minus 1] 119903

120581minus1]

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

le2119903R119890 Ψ (119911)

(1 minus 119903)2

(26)


119866 (119911) equiv120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

+120572[119865(119911)]

(120485)

[119911120573](120485)

minus 120574 (27)

and then119911Ψ1015840 (119911)

Ψ (119911)=

suminfin

120581=2120599 (120572 120591 120574 120573) [120581 minus 1] 119886

120581119911120581minus1

1 + suminfin

120581=2120599 (120572 120591 120574 120573) 119886

120581119911120581minus1

(28)

where

120599 (120572 120591 120574 120573) =120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591 minus 120574) [120573

119875(120485)

]

(29)

From (23) and (28) we got100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [120581 minus 1] 119903

120581minus1

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

(120581 ge 2) (30)


119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 Ψ (119911)

(1 minus 119903)2

(31)

Sinceinfin

sum120581=2


lt 2119903R119890 Ψ (119911)

1 +

infin

sum120581=2

2 (120573 minus 120485) 119903120581minus1

gt (1 minus 119903)2

(32)




(120485)

(120572 120591 120574 120573)



1199111198661015840 (119911)

119866 (119911) ≺ ℎ (119911) (33)

then

119866 (119911) ≺ 119902 (119911) = (120572 + 120591 minus 120574) exp [119899minus1

int119911

0

ℎ (120585) 120585minus1

119889120585 ] (34)



119866 (119911) = (120572 + 120591 minus 120574) +

infin

sum120581=2

120582120581[ [120573+120581minus1]119875

120485+1] 119886120581119911120581minus1

120573119875120485+1

(35)


1199111198661015840

(119911)

119866 (119911) ≺ ℎ (119911) (36)




1199111198661015840 (119911)

119866 (119911) ≺

radic119911

1 minus 119911(37)

then

119866 (119911) ≺ (120572 + 120591 minus 120574) 1 + radic119911

1 minus radic119911

1119899

120572 + 120591 = 120574 (38)




120573 gt 120485 + 1 with 120572 + 120591 = 0 If

119866 (119911) ≺ (120572 + 120591)1 + 119860119911

1 + 119861119911 (39)

then[119865(119911)]

(120485)

[119911120573](120485)

≺(120572 + 120591) (120573 minus 120485)

120591120581119911(120572+120591)(120485minus120573)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(40)


1

(120572 + 120591)119866 (119911) = 1 +

infin

sum120581=2

119862120581119886120581119911120581minus1

(41)

where

119862120581=

120582120581[ (120573+120581minus1)119875

120485+1] sdot [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591) [120573

119875120485+1

] (42)

Let 119892(119911) = [119865(119911)](120485)

[119911120573](120485) then 119892(119911) = 1 + sum

infin

120581=2120603120581119886120581119911120581minus1

where 120603120581= 120582120581[(120573+120581minus1)

119875120485+1

][(120573)

119875120485] and from (39) we have

119892 (119911) +120591

(120572 + 120591) (120573 minus 120485)1199111198921015840

(119911) ≺ (1 + 119860119911

1 + 119861119911) = ℎ (119911) (43)


119892 (119911) ≺(120572 + 120591) (120573 minus 120485)

120591120581119911minus(120572+120591)(120573minus120485)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(44)


Corollary 16 Let

119876(0)

(120572 120591 0 120573) =120572 (119865 (119911))

119911120573+

120591[119865 (119911)]1015840

[119911120573]1015840

(45)

If

119876(0)

(120572 120591 0 120573) ≺ (120572 + 120591) (1 + 119860119911

1 + 119861119911) (46)

then

119865 (119911)

119911120573≺

120572120573 + 120591 (120573 minus 120581)

120591120581119911minus120573(120572+120591)120591120581

times int119911

0

120585(120573(120572+120591)minus120591120581)120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(47)






Acknowledgments



References





















1 Introduction


119891 (119911) =119892 (119911)

ℎ (119911) (1)






R ℎ (119911) gt 0 (119911 isin 119880lowast

) (2)


119891 ≺ 119892 in 119880 or 119891 (119911) ≺ 119892 (119911) (119911 isin 119880lowast

) (3)


|119908 (119911)| le |119911| 119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880lowast

) (4)


119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0)

119891 (119880) sube 119892 (119880) (119911 isin 119880lowast

) (5)





2 Preliminaries


119891 (119911) =1

119911+

infin

sum119899=1

119886119899119911119899

(6)




119891119895(119911) =

1

119911+

infin

sum119899=1

119886119899119895119911119899

(7)


and 1198912(119911) by

(1198911lowast 1198912) =

1

119911+

infin

sum119899=1

11988611989911198861198992119911119899

(8)


119902120582120583

(119911) =1

119911+

infin

sum119899=1

(120582

119899 + 1 + 120582)

120583

119911119899

(120582 gt 0 120583 ge 0) (9)



119871120582120583119891 (119911) = (119891 (119911) lowast 119902

120582120583(119911))

=1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(10)


119905119886119892 and 119871


tively by

119871120582120583119892 (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198871198991003816100381610038161003816 119911119899


119871120582120583ℎ (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198881198991003816100381610038161003816 119911119899


ℎ (119911) isin Σ)

(11)


119871120582120583

(119891 lowast 119892) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast)

119871120582120583

(119891 lowast ℎ) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198881198991003816100381610038161003816 119911119899


)

(12)






property

120588119871120582120583

(119891 lowast 119892) (119911)

119871120582120583

(119891 lowast ℎ) (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911 (119911 isin 119880

lowast

) (13)


and 119871(120582 120583)(119891 lowast ℎ)(119911) = 0


120601 (120572 120573 119911) =1

119911+

infin

sum119899=0

(120572)119899+1

(120573)119899+1

119911119899

(14)



120601 (120572 120573 119911) =1

119911 21198651(1 120572 120573 119911) (15)

where

21198651(119887 120572 120573 119911) =

infin

sum119899=0

(119887)119899(120572)119899

(120573)119899

119911119899

119899(16)


120582120583(119911)


119871 (120572 120573 120582 120583) 119891 (119911) = (119891 (119911) lowast 120601 (120572 120573 119911) lowast 119902120582120583

(119911))

=1

119911+

infin

sum119899=1

(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(17)




119911(119871 (120572 120573 120582 120583) 119891 (119911))1015840

= 120572119871 (120572 + 1 120573 120582 120583) 119891 (119911)

minus (120572 + 1) 119871 (120572 120573 120582 120583) 119891 (119911) (18)


120588119871 (120572 + 1 120573 120582 120583) 119891 (119911)

119871 (120572 120573 120582 120583) 119891 (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911

(0 le 119861 lt 119860 le 1 120588 gt 0)

(19)



relation (19) above






infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861)

minus10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(20)


119891119899(119911) =

1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

(21)



1003816100381610038161003816100381610038161003816100381610038161003816

120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899) 119911119899+1

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899119911119899+1

)

minus11003816100381610038161003816100381610038161003816100381610038161003816

le 120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899)10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899

10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816)

minus1

le 1

(22)





100381610038161003816100381610038161003816100381610038161003816lt 1

(119911 isin 119880lowast)

(23)


Σ120583




120582(120588 119860 119861) then

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119899 ge 1

(24)


(21)


120582(120588 119860 119861)



1

119903minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903

le1003816100381610038161003816119891 (119911)

1003816100381610038161003816

le1

119903+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903


1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

le100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(25)



inequalityinfin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(26)


1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

le1

119903+ 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

119903+ 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|minus

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

ge1

119903minus 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

ge1

119903minus 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(27)


sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(119860 minus 119861) (2 + 120582)

120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(28)

Hence

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2+

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

le1

1199032+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

ge1

1199032minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816

ge1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(29)



120582(120588 119860 119861) which




disk |119911| lt 1199031 where

1199031= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(30)



119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (31)

For |119911| lt 1199031 we have

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 1 + 120582))

120583

119886119899119911119899

1119911 + suminfin

119899=1(120582(119899 + 1 + 120582))

120583

119886119899119911119899

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

100381610038161003816100381610038161003816100381610038161003816

lesuminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(32)


sum119899=1

(119899 + 1) (120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1


infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

)

(33)

or

suminfin

119899=1(119899 + 2 minus 120575) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(1 minus 120575)le 1 (34)


(119899 + 2 minus 120575) (120582(119899 + 1 + 120582))120583

|119911|119899+1

(1 minus 120575)

le (120582

119899 + 120582 + 1)

120583

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899 ge 1

(35)



|119911| le (1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899+1

(36)




disk |119911| lt 1199032 where

1199032= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(37)



1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891 (119911))10158401015840

(119891 (119911))1015840+ 2

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (38)






infin

sum119899=1

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(39)


119891119899(119911)

=1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

119899 ge 1

(40)




120582(120588 120572 120573 119860 119861) then1003816100381610038161003816119886119899

1003816100381610038161003816

le(119860 minus 119861) (119899 + 120582 + 1)

120583

(120573)119899+1

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861)) (120572)

119899+1

119899 ge 1

(41)


(40)




1199033= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(42)





1199034= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(43)






Acknowledgment


References






















1 Introduction


Φ (119911 119904 119886) =

infin

sum119899=0

119911119899

(119899 + 119886)119904


0 119904 isin C when |119911| lt 1

R (119904) gt 1 when |119911| = 1)

(1)


119904(120585) defined by

120577 (119904) =

infin

sum119899=1

1

119899119904= Φ (1 119904 1) = 120577 (119904 1) (R (119904) gt 1)

120577 (119904 119886) =

infin

sum119899=0

1

(119899 + 119886)119904


0)

ℓ119904(120585) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 1)119904= Φ (e2120587119894120585 119904 1) (R (119904) gt 1 120585 isin R)

(2)


119904(119911)

Li119904(119911) =

infin

sum119899=1

119911119899

119899119904= 119911Φ (119911 119904 1)


(3)


120601 (120585 119886 119904) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 119886)119904= Φ (e2120587119894120585 119904 119886)



when 120585 isin Z)

(4)




Φ (119911 119904 119886) =1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

1 minus 119911eminus119905d119905

(R (119886) gt 0R (119904) gt 0 when |119911| ≦ 1 (119911 = 1) R (119904) gt 1

when 119911 = 1) (5)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) =

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

119911119899

(119899 + 119886)119904

(119901 119902 isin N0 120582

119895isin C (119895 = 1 119901)

119886 120583119895isin C Z

minus

0(119895 = 1 119902)

120588119895 120590

119896isin R

+

(119895 = 1 119901 119896 = 1 119902)





lowast

)

(6)

with

nablalowast

= (

119901

prod119895=1

120588minus120588119895

119895) sdot (

119902

prod119895=1

120590120590119895

119895)

Δ =

119902

sum119895=1

120590119895minus

119901

sum119895=1

120588119895

Ξ = 119904 +

119902

sum119895=1

120583119895minus

119901

sum119895=1

120582119895+119901 minus 119902

2

(7)



(120582)120581=Γ (120582 + 120581)

Γ (120582)

=


1 (120581 = 0 120582 isin C 0)

(8)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0)

(9)



119898or

119897Ψ119898(119897 119898 isin N

0) involved


119897119865119898(119897 119898 isin N




1198871 119887

119898such that

119886119895isin C (119895 = 1 119897) 119887

119895isin C Z

minus

0(119895 = 1 119898)

(10)


119897Ψlowast

119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

=

infin

sum119899=0

(1198861)1198601119899sdot sdot sdot (119886

119897)119860119897119899

(1198871)1198611119899sdot sdot sdot (119887

119898)119861119898119899

119911119899

119899

=Γ (119887


119898)

Γ (1198861) sdot sdot sdot Γ (119886

119897) 119897Ψ119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

(119860119895gt 0 (119895 = 1 119897) 119861

119895gt 0 (119895 = 1 119898)

1 +

119898

sum119895=1

119861119895minus

119897

sum119895=1

119860119895≧ 0)

(11)


|119911| lt nabla = (

119897

prod119895=1

119860minus119860119895

119895) sdot (

119898

prod119895=1

119861119861119895

119895) (12)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

Γ (119904)intinfin

0


119905120582)

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0)

(13)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 0 120582)

= Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

= e119887Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 0)

(14)


119901 minus 1 = 119902 = 0 (1205821= 120583 120588

1= 1) (15)


Θ120582

120583(119911 119904 119886 119887)

=1

Γ (119904)intinfin

0


119905120582) (1 minus 119911eminus119905)

minus120583

d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0 120583 isin C)

(16)


R (119904) gt 0 when 119887 = 0 |119911| ≦ 1 (119911 = 1) (17)

or

R (119904 minus 120583) gt 0 when 119887 = 0 119911 = 1 (18)




Θ120582


Θ120582

120583(119911 119904 119886 0) = Φ

lowast

120583(119911 119904 119886) =

1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

(1 minus 119911eminus119905)120583d119905

(R (119886) gt 0 R (119904) gt 0 when |119911| ≦ 1 (119911 = 1)

R (119904 minus 120583) gt 0 when 119911 = 1)

(19)



[4] as follows

Φlowast

120583(119911 119904 119886) =

infin

sum119899=0

(120583)119899

(119886 + 119899)119904

119911119899

119899 (20)


120583(119911 119904 119886 119887) that is


Θ1

1(1 119904 119886 119887)

= 120577119887(119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1


119905) d119905

(21)




(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899

119899(120582 gt 0)

(22)



119867119898119899

pq (119911)

= 119867119898119899

pq [119911 |(119886

1 119860

1) (119886p 119860p)

(1198871 119861

1) (119887q 119861q)

]

=1

2120587119894intL

Ξ (119904) 119911minus119904d119904 (119911 isin C 0

1003816100381610038161003816arg (119911)1003816100381610038161003816 lt 120587)

(23)

where

Ξ (119904) =prod

119898

119895=1Γ (119887

119895+ 119861

119895119904) sdot prod

119899

119895=1Γ (1 minus 119886

119895minus 119860

119895119904)

prodp

119895=119899+1Γ (119886

119895+ 119860

119895119904) sdot prod

q

119895=119898+1Γ (1 minus 119887

119895minus 119861

119895119904)

(24)


119895gt 0 (119895 =

1 p) 119861119895gt 0 (119895 = 1 q) 119886

119895isin C (119895 = 1 p)



Γ (119887119895+ 119861

119895119904)

119898

119895=1

(25)


Γ (1 minus 119886119895+ 119860

119895119904)

119899

119895=1

(26)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902





1 minus 119905)

=

infin

sum119899=0

119871(120572)

119899(119909) 119905

119899

(|119905| lt 1 120572 isin C)

(27)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=eminus119887

Γ (119904)

infin

sum119899=0

119899

sum119896=0

(minus1)119896

(119899

119896) sdot Γ (119904 + 120582 (120572 + 119896 + 1))

sdot 119871(120572)

119899(119887)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 + 120582 (120572 + 119895 + 1) 119886)

(R (119886) gt 0 R (119904 + 120582120572) gt minus120582)

(28)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) (29)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



D120572

119911119911

119901

119891 (119911) =1

Γ (minus120572)int119911

0

119891 (120585) 120585119901


(R (120572) lt 0 R (119901) gt 1)

(30)


D120572

119911119911

119901

119891 (119911) =d119898

d119911119898D

120572minus119898

119911119911

119901

119891 (119911) (31)



Im(120585)

Branch line for

Branch line for


exp[p(ln(g(120585)))]

C1

C2

C3

C4

Re(120585)

z

a

gminus1(0)





119901


119863120572

119892(119911)[119892 (119911)]

119901

119891 (119911)

=eminus119894120587119901Γ (1 + 120572)4120587 sin (120587119901)

times int119862(119911+119892

minus1(0)+119911minus119892

minus1(0)minus119865(119886)119865(119886))

sdot119891 (120585) [119892 (120585)]

119901

1198921015840 (120585)

[119892 (120585) minus 119892 (119911)]120572+1

d120585

(32)



1cup119862

2cup119862

3cup119862








119892(119911)[119892(119911)]

119901



119863120572

119911119911

119901

=Γ (1 + 119901)

Γ (1 + 119901 minus 120572)119911119901minus120572

(R (119901) gt minus1) (33)



11205881199011205901120590119902)

12058211205821199011205831120583119902


119863120572

119911119911

120573minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899)

1198871120582

| (119904 1) (01

120582)]119863

120572

119911119911

120573minus1119911119899

119899

=Γ (120573)

120582Γ (119904) Γ (120573 minus 120572)

infin

sum119899=0

(120573)119899

(120573 minus 120572)119899

sdot

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899+120573minus120572minus1

119899

=Γ (120573)


Φ(1205881120588119901112059011205901199021)

12058211205821199011205731205831120583119902120573minus120572

(119911 119904 119886 119887 120582)







119863120572

119892(119911)119891 (119911) = 119863

120572

ℎ(119911)119891 (119911) 119892

1015840(119911)

ℎ1015840(119911)(ℎ(119908) minus ℎ(119911)

119892(119908) minus 119892(119911))

120572+1

100381610038161003816100381610038161003816100381610038161003816119908=119911

(35)




119911119901119874120572

120573119891 (119911)

= 119901(119911119901minus1

)minus120572

sdot119911

119874120572

120573119891 (119911) (119911

119901minus1

)120572

119901minus1

prod119904=1

(1 minus119911

119908eminus(2120587119894119904)119901)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(36)

where

119892(119911)119874120572


Γ (120573)

Γ (120572)[119892 (119911)]

1minus120573

119863120572minus120573

119892(119911)[119892 (119911)]

120572minus1

sdot sdot sdot

(37)



120579 (119911) = (119911 minus 1199110) 119902 (119911) (38)


119870

=0 1 [119888] ([119888] 119905ℎ119890 119897119886119903119892119890119904119905 119894119899119905119890119892119890119903 119899119900119905 119892119903119890119886119905119890119903 119905ℎ119886119899 119888)

(39)


0and let

120596 = exp (2120587119894119886) (40)



sum119896isin119870

119888minus1

120596minus120574119896

119891 (120579minus1

(120579 (119911) 120596119896

))

=

infin


[120579 (119911)]119888119899+120574

Γ (119888119899 + 120574 + 1)

sdot119863119888119899+120574

119911minus119887119891 (119911) 120579

1015840

(119911) (119911minus119911

0

120579 (119911))

119888119899+120574+1

100381610038161003816100381610038161003816100381610038161003816119911=1199110

(1003816100381610038161003816119911minus1199110

1003816100381610038161003816=100381610038161003816100381611991101003816100381610038161003816)

(41)



119891 (119911) = 119888

infin


(119911 minus 1199110)119888119899+120574

Γ(119888119899 + 120574 + 1)119863

119888119899+120574

119911minus119887119891(119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199110

(42)



1)(119911minus119911

2)) where 119911

1and 119911

2are two



120596 = exp(2120587119894119886) (43)


1and 119911


curves

119862 (119905) 119862 (119905) subR 0 lt 119905 ≦ 119903 (44)

be defined by

119862 (119905) = 1198621(119905) cup 119862

2(119905)

= 119911 1003816100381610038161003816120582119905 (1199111 1199112 119911)

1003816100381610038161003816 =1003816100381610038161003816100381610038161003816120582119905(119911

1 119911

21199111+ 119911

2

2)1003816100381610038161003816100381610038161003816

(45)

where

120582119905(119911

1 119911

2 119911) = [119911 minus

1199111+ 119911

2

2+ 119905 (

1199111minus 119911

2

2)]

sdot [119911 minus (1199111+ 119911

2

2) minus 119905 (

1199111minus 119911

2

2)]

(46)


1+ 119911



1199111+ 119911

2

2+ (

1199111minus 119911

2

2) 119905 (47)


2

2minus (

1199111minus 119911

2

2) 119905 (48)

Im(120585)

Branch line for

Branch line for

C1

C2

z1

z2

Re(120585)

z

c =z1 + z2

2

(120585 minus z2)120583120579(120585)

aminus120574minus1

(120585 minus z1)120579(120585)

aminus120574minus1



[(119911 minus 1199111) (119911 minus 119911

2)]

120582


2))))

(49)


plusmn

defined by

119871plusmn=

119911 119911 =1199111+ 119911

2

2plusmn 119905 (

1199111minus 119911

2

2) (0 ≦ 119905 ≦ 1)

119911 119911 =1199111+ 119911

2

2plusmn 119894119905 (

1199111minus 119911

2

2) (119905 lt 0)

(50)



1)(119911minus119911

2) gt 0 (v)


2)119901


+cup 119871


119911minus1199112

(119911 minus 1199112)119901


119870 = 119896 119896 isin N arg(120582119905(119911

1 119911

21199111+ 119911

2

2))

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) +

2120587119896

119886

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) + 2120587

(51)


defined by

120585 =1199111+ 119911

2

2+1199111minus 119911

2


sum119896isin119870

119888minus1120596minus120574119896

1199111minus 119911

2

119891 (120601minus1

(120596119896

120601 (119911)))

times [120601minus1

(120596119896

120601(119911)) minus 1199111]][120601

minus1

(120596119896

120601(119911)) minus 1199112]120583

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

[120601 (119911)]119888119899+120574

(52)


where

120601 (119911) =119911 minus 119911

1

119911 minus 1199112

(53)



2)120583

1199111minus 119911

2

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

(119911 minus 119911

1

119911 minus 1199112

)

119888119899+120574

(54)


2) where 119911

1and 119911




sum119896isin119870

119888minus1

120596minus120574119896

[(1199112minus 119911

1+ radicΔ

119896

2)

120572

(1199111minus 119911

2+ radicΔ

119896

2)

120573

sdot 119891 (1199111+ 119911

2+ radicΔ

119896

2)


sdot (1199112minus 119911

1minus radicΔ

119896

2)

120572


2minus radicΔ

119896

2)

120573

119891(1199111+ 119911

2minus radicΔ

119896

2)]

=

infin



eminus119894120587119888(119899+1)[120579 (119911)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911minus1199112


(120579 (119911)

(119911minus1199112) (119911minus119911

1))


1205791015840

(119911)119891 (119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199111

(55)

where

Δ119896= (119911

1minus 119911

2)2

+ 4119881 (120596119896

120579 (119911))

119881 (119911) =

infin

sum119903=1

119863119903minus1

119911[119902 (119911)]

minus119903

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

119911119903

119903

120579 (119911) = (119911 minus 1199111) (119911 minus 119911

2) 119902 ((119911 minus 119911

1) (119911 minus 119911

2))

(56)


(119911 minus 1199111)(119911 minus 119911

2)) and 119911


119891 (119911) = 119888119911minus120573

(119911 minus 1199111)minus120572

times

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot119863minus120572+119888119899+120574

119911119911


(119911 + 119908 minus 1199111)119891(119911)

10038161003816100381610038161003816119911=1199111

(119908=119911)

(57)



119863120572

119911119911

119901+119902

119906 (119911) V (119911)

=

infin


(120572

120574 + 119899)119863

120572minus120574minus119899

119911119911

119901

119906 (119911)119863120574+119899

119911119911

119902V (119911) (58)




119863120572

119911119911

120572+120573minus1

119906 (119911) V (119911)


sdot

infin


119863120572+]+1minus119899119911



(60)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 (1120582))]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot 119865(119896minus1)



where 120582 gt 0 and 119865(119899)119863


119865(119899)

119863[119886 119887

1 119887

119899 119888 119909

1 119909

119899]

=

infin

sum1198981119898119899=0

(119886)1198981+sdotsdotsdot+119898

119899

(1198871)1198981

sdot sdot sdot (119887119899)119898119899

(119888)1198981+sdotsdotsdot+119898

119899

1199091198981

1


119909119898119899

119899

119898119899

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091198991003816100381610038161003816 lt 1)

(62)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119896

(119911119896minus1)120572

sdot119911119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) (119911119896minus1

)120572

sdot

119896minus1

prod119904=1

(1 minus119911

119908eminus2120587119894119904119896)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(63)




119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

(64)



119896

(119911119896minus1)120572

119911

119874120572

120573

times

1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119911119899+(119896minus1)120572

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


sdot((119911119908) eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119908=119911

=119896

120582Γ (119904) (119911119896minus1)120572

times

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981



sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

times119911

119874120572

120573119911


119896minus1+119899+(119896minus1)120572

=119896

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


times(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

sdotΓ (120573) Γ (119898


119896minus1+ 119899 + 119896120572)

Γ (120572) Γ (120573 + 1198981+ sdot sdot sdot + 119898

119896minus1+ 119899 + 119896120572)


=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 1120582)]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(119899+119896120572)1198981+sdotsdotsdot+119898

119896minus1

(1+120572minus120573)1198981



119896minus1

sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

(65)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


119911minus1198881198990(119911 minus 119911

0)119888119899

Γ (119888119899 + 1) Γ (1 minus 119888119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(1003816100381610038161003816119911 minus 1199110

1003816100381610038161003816 =100381610038161003816100381611991101003816100381610038161003816 120582 gt 0)

(66)


Proof Setting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


0 we

have

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


(119911 minus 1199110)119888119899

Γ (1 + 119888119899)

sdot119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

(67)


0| = |119911

0|


119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

=119911minus1198881198990

Γ (1 minus 119888119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(68)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(69)


119911 =1199111

2+1199111



Proof By taking 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 9 with 119911

2= 0 120583 = 120572 ] = 120573 and 0 lt 119888 ≦ 1 we find

that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888(119911 minus 1199111)minus120573

119911minus120572

1199111

sdot

infin



sdot (119911 minus 119911

1

119911)119888119899+120574

sdot119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

(71)



119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120572+120573minus1

1

Γ (120572 + 119888119899 + 120574)

Γ (120572 + 120573)

times Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582)

(72)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573+120574

(119911 minus 1199111)minus120572+120574

119911120573+120572minus2120574minus1

1

sdot

infin


sin [(120573 minus 119888119899 minus 120574) 120587] e119894120587119888(119899+1)

sin [(120573 + 119888 minus 120574) 120587] Γ (1 minus 120572 + 119888119899 + 120574)

sdot (119911(119911 minus 119911

1)

11991121

)

119888119899

Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdot [(119911 minus 1199111)Φ

(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (120573 minus 119888119899 minus 120574

120572 + 120573 minus 2119888119899 minus 2120574) 119911

1

sdotΦ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)]

(73)


119911 =1199111

2+1199111



Proof Putting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 10 with 119911

2= 0 0 lt 119888 ≦ 1 119902(119911) = 1 and 120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2) we find that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573

(119911 minus 1199111)minus120572

sdot

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

(75)


119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

= 119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

+ (119911 minus 1199111)119863

minus120572+119888119899+120574

119911


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120573+120572minus2119888119899minus2120574

1


Γ (1 + 120573 + 120572 minus 2119888119899 minus 2120574)

sdot Φ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (119911 minus 119911

1

1199111

)Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdotΦ(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582) )

(76)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

=Γ (120591) Γ (1 + ] minus 120591) sin (120574120587)

120587

sdot

infin


(minus1)119899

Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)


(77)



12058211205821199011205831120583119902


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=

infin


(] minus 120591120574 + 119899


119911]minus1

sdot 119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(78)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)



119911]minus1 =

Γ (])Γ (120591 + 120574 + 119899)

119911120591+120574+119899minus1

119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911minus120574minus119899

Γ (1 minus 120574 minus 119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(79)



Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

= (Γ (120591) Γ (1 + ] minus 120591) sin (120573120587)




sdotsin (120579120587)

sin [(120579 + 120574) 120587]

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(80)



120572 = ] minus 120591 119906 (119911) = 119911120591minus120573

V (119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(81)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


sdot

infin


119863]minus120591+120574+1minus119899119911




119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(82)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

119863]minus120591+120574+1minus119899119911


=Γ (] minus 120579 minus 119899)


119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (120579 + 119899)

Γ (1 + 120579 + 120574)

times 119911120579+120574

Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(83)





lim119887rarr0

11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)] = 120582Γ (119904)

(120582 gt 0)

(84)



Φ(120588112058811990113120590

112059011990213)

12058211205821199011205721205831120583119902120573

(119911 119904 119886)

=3Γ (120573) Γ (3120572)

Γ (120572) Γ (120573 + 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

(3120572)119899

(120573 + 120572)119899

119911119899

(119899 + 119886)119904


(85)


1198651[119886 119887

1 119887

2 119888 119909

1 119909

2]

=

infin

sum11989811198982=0

(119886)1198981+1198982

(1198871)1198981

(1198872)1198982

(119888)1198981+1198982

1199091198981

1

1198981

1199091198982

2

1198982

(max 100381610038161003816100381611990911003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 lt 1)

(86)




1= 120583 in



Θ120582

120583(119911 119904 119886 119887)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(111)

120583120572+119888119899+120574120572+120573(119911

1 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(87)


119911 =1199111

2+1199111





Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886)

=Γ (120591) Γ (1 + ] minus 120591)



times (Γ (1 + 120574 + 120579) sin [(] minus 120591 + 120573) 120587]


sdot

infin


Γ (]minus120579minus119899) Γ (120579+119899)Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120582

(119911 119904 119886)





References












































Operations Research








Contents





















Toka Diagana










1 Introduction




119886(119899 119896) and 119878



(119905)119899=

119899

sum119896=0

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

(119905 minus 119886)119896

(1)

(119905 minus 119886)119899

=

119899

sum119896=0

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(119905)119896 (2)


119904119886(119899 119896) =

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

119878119886(119899 119896) =

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(3)


119904119886(119899 + 1 119896) = 119904

119886(119899 119896 minus 1) + (119886 minus 119899) 119904

119886(119899 119896) (4)

119878119886(119899 + 1 119896) = 119878

119886(119899 119896 minus 1) + (119896 minus 119886) 119878

119886(119899 119896) (5)


119904119886(0 0) = 1 119904

119886(119899 0) = (119886)

119899 119904

119886(0 119896) = 0 119899 119896 = 0

119878119886(0 0) = 1 119878

119886(119899 0) = (minus119886)

119899

119878119886(0 119896) = 0 119899 119896 = 0

(6)


1+ 119883


119896of





119875 [119882 = 119899] =119896

(1 minus 120579)minus119904

(minus log (1 minus 120579))119896

120579119899

119899(minus1)

119899minus119896

119904minus119904(119899 119896)

119875 [119885 = 119899] =119896

119890119898 (119890119897 minus 1)119896

119897119899

119899(minus1)

119899minus119896

119878minus119898119897

(119899 119896)

(7)






119870(1)

119899119903= 119903 + [log(119899 minus 1


1

119903+ 119900 (1)] (8)


2 Explicit Formula



119865119899(119909) = prod

119899


1198651015840

119899(119909) = sum


119899minus1le119899

119894isinN11989911989511198952119895119899minus1

1198911015840

119894(119909)

119899minus1

prod119896=1

119891119895119896(119909) (9)


1 2

119889

119889120585(120585119895 + 119886)

1= 119895

119889

119889120585(120585119895 + 119886)

2= (120585119895 + 119886) 119895 + (120585119895 + 119886 minus 1) 119895

(10)


1

119895119899119889

119889120585(120585119895 + 119886)



119899minus1

prod119896=1

(120585 +119886 minus 119895

119896

119895) (11)



1

119895119899119889119896

119889120585119896(120585119895 + 119886)


119899minus119896le119899minus1

119896

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(12)


1

119895119899119889119898

119889120585119898(120585119895 + 119886)


119899minus119898le119899minus1

119898

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(13)

Then

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899


119899minus119898le119899minus1

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(14)




119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895) = sum



prod119902=1

(120585 +119886 minus 119894

119902

119895)

(15)

119894119902

isin 1198951 1198952 119895

119899minus119898 has ( 119899minus119898





119902=1(120585 + (119886 minus





119895119899minus119898minus1





119902=1(120585 + (119886 minus 119895


of (1119895119899)(119889119898+1119889120585119898+1)(120585119895 + 119886)119899 Thus we have

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899



119898 + 1

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)



119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(16)

Lemma 2 Let 119904(119899 119896 119886) =

(1119896)lim120585rarr0

((sum119896

119895=0(minus1)

119896minus119895

( 119896119895) (119889119896119889120585119896)(120585119895 + 119886)

119899)119896)

Then


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (17)

Proof Using Lemma 1

119904 (119899 119896 119886)

=1

119896lim120585rarr0

((

119896

sum119895=0


(119896

119895) 119895119899


119899minus119896le119899minus1

119896

times

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)) (119896)

minus1

)

(18)



(120585 + (119886 minus 119895119902)119895) = (1119895119899minus119896)prod

119899minus119896

119902=1(120585119895 + 119886 minus 119895

119902)


1

119896lim120585rarr0

[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(120585119895 + 119886 minus 119895119902)]

]

=1

119896[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)]

]

(19)

which boils down to


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (20)

since

1

119896

119896

sum119895=0


(119896

119895) 119895119896

= 119878 (119896 119896) = 1 (21)



119904119886(119899 119896) = 119904 (119899 119896 119886) = sum


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (22)

Proof We know that


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (23)



119899+1minus119896)







119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (24)

is equal to


119899minus119896+1le119899minus1

119899minus119896+1

prod119902=1

(119886 minus 119895119902) + (119886 minus 119899)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)

(25)

This implies that

119904 (119899 + 1 119896 119886) = 119904 (119899 119896 minus 1 119886) + (119886 minus 119899) 119904 (119899 119896 119886) (26)



119886(119899 119896)


120582 = (1205821ge 120582


119896) (27)


119894


14= 11989115

= 11989123

= 11989131

= 11989142

= 1 119891119894119895= 0 elsewhere

(1 le 119895 le 120582119894)

0 0 0 1 10 0 11 0 00 10

(28)



(119886119894)119894ge0



1198860 1198861 119886

ℎ then |119879119889119860(ℎ 119903)| = ( ℎ+1




120596-tableau is











119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)


120596 (|119888|) (29)




is equal to 119904119886(119899 119896)



0 1 2 119899 minus 1 and

120596119860(120601) = prod

119888isin120601

120596 (|119888|)

=

119899minus119896

prod119894=1

(119886 minus 119895119894) 119895

119894isin 0 1 2 119899 minus 1

(30)

If 119886 = 1198861+ 1198862for some 119886

1and 119886

2 then with 120596lowast(119895) = 119886

2minus 119895

120596119860(120601) =

119899minus119896

prod119894=1

(1198861+ 120596lowast

(119895119894))

=

119899minus119896

sum119903=0

119886119899minus119896minus119903

1sum


119903le119895119899minus119896

119903

prod119894=1

120596lowast

(119902119894)

(31)






may write

120596119860(120601) = sum

120595isin119861120601

120596119860(120595) (32)


120596119860(120595) = 119886

119899minus119896minus119903

1

119903

prod119894=1

120596lowast

(119902119894) (33)

where 1199021 1199022 119902

119903isin 119895

1 1198952 119895


written as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

sum120595isin119861120601

120596119860(120595) (34)



1 1198952 119895

119899minus119896



(119899

119896)(

119899 minus 119896

119903) (35)





( 119899119896) ( 119899minus119896

119903)

( 119899119903)

= (119899 minus 119903

119896) (36)


119904119886(119899 119896) =

119899minus119896

sum119903=0

(119899 minus 119903

119896) 119886119899minus119896minus119903

1sum120595isin119861119903


120596lowast

(|119888|) (37)



119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

1sum



120596lowast

(|119888|) (38)

Clearly 119861119899minus119895



tity

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

11199041198862

(119899 119895) (39)


1and 119886

2





119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (40)






2isin 1198791198891198602(119895 minus 1 119895 minus 119899+ 119896)


2 we obtain an


119860

(119898 +


sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601)

=

119899

sum119896=0

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011)

times

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

(41)


Note that

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)


119895minus119899+119896le119898+119895minus1

119895minus119899+119896

prod119902=1

(119886 minus 119892119902)



119895minus119899+119896

prod119902=1

(119886 minus (119898 + 119892119902))

= 119904119886minus119898

(119895 119899 minus 119896)

(42)


sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011) = 119904

119886(119898 119896)

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601) = 119904

119886(119898 + 119895 119899)

(43)

Thus

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (44)



convolution formula

119904119886(119899 + 1119898 + 119895 + 1) =

119899

sum119896=0

119904119886(119896 119899) 119904

119886minus(119896+1)(119899 minus 119896 119895) (45)

Proof Let


in 1198601= 0 1 119896 minus 1


are in 1198602= 119896 + 1 119899








21199092 + sdot sdot sdot + 119886

119899119909119899 has only

real zeros then

1198862

119896ge 119886119896+1

119886119896minus1

119896

119896 minus 1

119899 minus 119896 + 1

119899 minus 119896(119896 = 2 119899 minus 1) (46)


sum119896=0

119904119886(119899 119896) (119905 + 119886)

119896

(47)



119896=0119904119886(119899 119896)119905119896 are 119886 119886minus1 119886minus



119899


and hence unimodal


⟨119905⟩119899=

119899

sum119896=0


119904119886(119899 119896) (119905 + 119886)

119896

(48)



119904119886(119899 119896) = (minus1)

119899minus119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902) (49)


119886(119899 119896)| as

1003816100381610038161003816119904119886 (119899 119896)1003816100381610038161003816 = (minus1)

119899minus119896

119904119886(119899 119896)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902)

(50)



lt 1199062


infin

sum119894=1

1

119906119894

= infin

infin

sum119894=1

1

1199062119894

lt infin (51)




which sum119899119896


119870119899= 119899 minus [

119899

sum119894=1

1

119906119894

minus

119899

sum119894=1

1

1199062119894

(1 +1

119906119894

)

minus1

+ 119900 (1)] (52)


1

1+1

2+ sdot sdot sdot +

1

119899= log 119899 + 120574 + 119900 (1) (53)



119886(119899 119896)|

119899

119896=0



119899


mation

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (54)




Denoting by 119896119886119899



1= 119887 + 0 119906

2= 119887 + 1 119906


119896119886119899

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894)2(1 +

1

119887 + 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894) (119887 + 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894 + 1+ 119900 (1)]

(55)


119899minus1

sum119894=0

1

119887 + 119894 + 1= log (119887 + 119899) minus log 119887 (56)

From this we get

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (57)



119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (58)



1)(119886 minus


119899minus119896)where 119895


1 By Theorem 10 119904119886(119899 119896) = sum

119899119899minus119896 Thus with 119906

1= 119886 119906

2=

119886 minus 1 119906119899minus1

= 119886 minus 119899 + 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894)2(1 +

1

119886 minus 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894) (119886 minus 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119886 minus 119894 + 1+ 119900 (1)]

(59)


119899119896 0 1 2 30 11 12 2 13 6 6 14 24 35 10 15 120 225 85 156 720 1624 735 1757 5040 13132 6769 19608 40320 109584 118124 672849 362880 1026576 1172700 72368010 3628800 10628640 12753576 8409500


119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (60)


9


119896minus19

= [log(1 + 8

1) + 119900 (1)]

= [log 9 + 119900 (1)]

asymp 2

(61)


10


1198961010

= [log( 10 + 1

10 minus 10 + 1) + 119900 (1)]

= [log 11 + 119900 (1)]

asymp 2

(62)







(63)


Table 2 Values of 11990410(119899 119896)

119899119896 0 1 2 30 11 10 12 90 19 13 720 242 27 14 5040 2414 431 345 30240 19524 5000 6356 151200 127860 44524 81757 604800 662640 305956 772248 1814400 2592720 1580508 5376289 3628800 6999840 5753736 265576410 3628800 10628640 12753576 8409500


119904(119899 119896) 0 1 2 3 4 50 11 0 12 0 1 13 0 2 3 14 0 6 11 6 15 0 24 50 35 10 16 0 120 274 225 85 157 0 720 1764 1624 735 1758 0 540 13068 13132 6769 19609 0 40320 109584 118124 67284 2244910 0 362880 1026576 1172700 723680 269325





2= 2 119906


119896119899= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

1

1198942(1 +

1

119894)minus1

+ 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

(1

119894minus

1

119894 + 1) + 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894 + 1+ 119900 (1)]

(64)


119899minus1

sum119894=1

1

1 + 119894= log 119899 minus log 1 + 120574 + 119900 (1) (65)

Therefore we have

119896119899= [log 119899 + 120574 + 119900 (1)] (66)



7

119896=0occurs at

1198967= [log 7 + 120574 + 119900 (1)]

= [195 + 05772 + 119900 (1)]

= [253 + 119900 (1)]

asymp 2

(67)


11989610

= [log 10 + 120574 + 119900 (1)]

= [230 + 05772 + 119900 (1)]

= [28772 + 119900 (1)]

asymp 3

(68)




Acknowledgment


References
















1 Introduction















2 New Instructions







x

f(x)

i

(ii) and (iii)1

c













2119888 and



















2minus119890


minus119873

2minus119873

] (1)









2119909 multiplica-








1sdot finite

1+ finite

2= 2 (2)








2120583 in the case






2((maximum



Inputsign exp mant

Drop leading zeros

numZeros + 1


Add

Switch

s e12 m51

lookup output


exp

Add 1

expP1

adjustedE

Switch

bitRot


Add 0x7ff

Add 0x3ff

Switch

Concatenate bit

xor

Add drop carry

moreThan22Zeros

isNot12

Concatenate bit

51

42

42 + 1

44

30

31

52

52

52

52


Add



(c)

(b)

(d)

(e)

(f)

(g)


(h)

(i)

(j)

(k)

leading12Mant

(a)

(p)

(r)

(o)

(l)

(m)

(n)

(q)

(s)

(t)

(u)

(v)

5

sim

sim

0x7ffSubs

Subs 0b0 0




12-bit lookup

12-bit lookup

ge1

e(last 10bits)lt1

sim


0b10 0Subs

Subs 0b0 0

low 10bits4

6

11

11

11

11

12

Lower 10






2119872

recip 3 113 376 0500 2048 3 infin






3 Evaluation





4 Conclusion




(a)


recip (2minus119890

119888)ext (2minus119890



119888)sat

sqrt (2minus119890

119888)ext 21198902

119888 0infin 0 NaN 0 0 0 NaN




(b)


minusinfin 119888 119888 0 0

infin 119888 0 119888 0

NaN 119888 119888 119888 0

(c)



minusinfin119891

minusinfin119891

infin infin119891

infin119891

infin119891

infin119891

NaN0

NaN119891

NaN119891

NaN119891

NaN119891

NaN1

2119891

2119891

2119891

2119891

NaN2

1radic2119891

1radic2119891

1radic2119891

1radic2119891

NaN3

0119891

0119891

0119891

0119891

recipdiv

sqrtrsqrtcbrt

rcbrtqdrt

rqdrtlog2

loglog1pexp2

expexpm1

atan2atanasinacos

sincostan

0 5 10 15 20 25 30







Acknowledgments


References

























1 Introduction








[119865 (119905)]119896

= sum119899gt0

119865 (119899 119896) 119905119899

(1)





119860Δ

(119899 119896) = 120572119896

119865Δ

(119899 119896) (2)



119860Δ

(119899 119896) = 120572119899

119865Δ

(119899 119896) (3)


119891(119899)119905119899 119877(119905) = sum119899gt0

119903(119899)119905119899


119860Δ

(119899 119896) =

119899

sum119898=119896

119865Δ

(119899119898)119866Δ

(119898 119896) (4)


119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899



119886(119899)119905119899

are

119886 (119899) =

119899

sum119896=1

119865Δ

(119899 119896) 119903 (119896)

119886 (0) = 119903 (0)

(5)




119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899


119899ge0119886(119899)119905119899 are

119886 (119899) =

119899

sum119896=0

119891 (119896) 119903 (119899 minus 119896) (6)




119910119899(119909) =

119899

sum119896=0

(119899 + 119896)

(119899 minus 119896)119896(119909

2)119896

(7)


119901119899(119909) = 119909

119899

119910119899minus1

(1

119909) (8)


119901119899(119909) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)(1

2)119899minus119896

119909119896 (9)


sum119899=0

119901119899(119909)

119899119905119899


(10)




119862Δ

(119899 119896) =119896

119899(2119899 minus 119896 minus 1

119899 minus 1) (11)



2)

119861 (119905) = 119890119905

=

infin

sum119899=0

1

119899119905119899

(12)


119860Δ

(119899 119896) = (2119909)119896

(1

2)119899

119862Δ

(119899 119896) = 2119896minus119899

119909119896119896

119899(2119899 minus 119896 minus 1

119899 minus 1)

(13)


119899=0119887(119899)119905119899

are equal to

119887 (119899) =1

119899 (14)


119901119899(119909) = 119899

119899

sum119896=1

119860Δ

(119899 119896) 119887 (119896) =

119899

sum119896=1

(2119899 minus 119896 minus 1)


119909119896

(15)




119898119899+1

(119909 120573 119888) =(119888 minus 1)

119899

119888119899

sdot ((119909 minus 119887119899)119898119899(119909 120573 119888) minus 120582

119899119898119899minus1

(119909 120573 119888))

(16)

where

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) = 119909 minus 119887

0

119887119899=

(1 + 119888) 119899 + 120573119888

1 minus 119888

120582119899=

119888119899 (119899 + 120573 minus 1)

(1 minus 119888)2

(17)


1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) =

119909 (119888 minus 1) + 120573119888

119888

1198982(119909 120573 119888)

=1199092 (119888 minus 1)

2

+ 119909 ((2120573 + 1) 1198882 minus 2120573119888 minus 1) + (120573 + 1) 1205731198882

1198882

(18)


infin

sum119899=0

119898119899(119909 120573 119888)

119899119905119899

= (1 minus119905

119888)119909


(19)





119860 (119905) = (1 minus119905

119888)119909

=

infin

sum119899=0

(119909

119899) (minus1)

119899

119888minus119899

119905119899


=

infin

sum119899=0


119899) (minus1)

119899

119905119899

(20)


suminfin

119899=0119886(119899)119905119899 and 119861(119905) = sum

infin


follows

119886 (119899) = (119909

119899) (minus1)

119899

119888minus119899

119887 (119899) = (minus119909 minus 120573

119899) (minus1)

119899

(21)


119898119899(119909 120573 119888) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896)

= (minus1)119899

119899

119899

sum119896=0

(119909

119896)(


119899 minus 119896) 119888minus119896

(22)



119872119899+1

(119909 120575 120578) = (119909 minus 119887119899)119872119899(119909 120575 120578) minus 120582

119899119872119899minus1

(119909 120575 120578)

(23)

where1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 119887

0

119887119899= (2119899 + 120578) 120575

120582119899= (1205752

+ 1) 119899 (119899 + 120578 minus 1)

(24)


1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 120575120578

1198722(119909 120575 120578) = 119909

2

minus 1199092120575 (1 + 120578) + 120578 ((120578 + 1) 1205752

minus 1)

(25)


infin

sum119899=0

119872119899(119909 120575 120578)

119899119905119899

= ((1 + 120575119905)2

+ 1199052

)minus1205782


1 + 120575119905))

(26)




119860 (119905) = ((1 + 120575119905)2

+ 1199052

)minus1205782

= (1 + 2120575119905 + (1205752

+ 1) 1199052

)minus1205782

119861 (119905) = exp (119909tanminus1 ( 119905

1 + 120575119905))

(27)


1(1198602(119905)) and we expand 119860


theorem

1198601(119905) = (1 + 119905)

minus1205782

=

infin

sum119899=0

(minus120578

2119899

) 119905119899

1198602(119905) = 2120575119905 + (120575

2

+ 1) 1199052

(28)


infin

119899=01198861(119899)119905119899

are

1198861(119899) = (

minus120578

2119899

) (29)


1198862119896minus119899

119887119899minus119896

(119896

119899 minus 119896) (30)


119860Δ

2(119899 119896) = (2120575)

2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896) (31)


infin

119899=0119886(119899)119905119899

119886 (0) = 1198861(0)

119886 (119899) =

119899

sum119896=1

119860Δ

2(119899 119896) 119886

1(119896)

(32)


119886 (0) = 1

119886 (119899) =

119899

sum119896=1

(2120575)2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896)(

minus120578

2119896

) (33)


1(1198612(119905)) where

1198611(119905) = 119890

119905

=

infin

sum119899=0

1

119899119905119899

1198612(119905) = 119909tanminus1 ( 119905

1 + 120575119905)

(34)



functions 1199091198621(1198622(119905)) where

1198621(119905) = tanminus1 (119905)

1198622(119905) =

119905

1 + 120575119905

(35)


((minus1)(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896] (36)

where [119895



119862Δ

1(119899 119896) = ((minus1)

(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

sdot

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896]

(37)


(119899 minus 1

119896 minus 1) 119886119896

119887119899minus119896

(38)


to

119862Δ

2(119899 119896) = (

119899 minus 1

119896 minus 1) (minus120575)

119899minus119896

(39)


2(119905)

119861Δ

2(119899 119896) = 119909

119896

119899

sum119898=119896

119862Δ

2(119899119898) 119862

Δ

1(119898 119896) (40)


infin

119899=01198871(119899)119905119899

are defined by

1198871(119899) =

1

119899 (41)


infin

119899=0119887(119899)119905119899

119887 (0) = 1198871(0)

119887 (119899) =

119899

sum119896=1

119861Δ

2(119899 119896) 119887

1(119896)

(42)


119887 (0) = 1

119887 (119899) =

119899

sum119896=1

119909119896

119899

119899

sum119898=119896

(119899

119898)120575119899minus119898

(1 + (minus1)

119898+119896

(minus1)(119898+119896)2minus119899

)

sdot

119898

sum119895=119896


(119895 minus 1)(119898

119895)[

119895

119896]

(43)


119872119899(119909 120575 120578) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896) (44)



Acknowledgment


References





































119892 (119911) =1

119911+ 1198870+

infin

sum119899=1

119887119899119911119899 (1)




1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911) (2)






119899| le

(119861 minus 119860)(119899 + 1) for 119892 isin Σ[119860 119861]



ℎ (119908) = 119892minus1

(119908) =1

119908+

infin

sum119899=0

119861119899119908119899

=1


1

119899119870119899

119899+1(1198870 1198871 119887

119899) 119908119899

(3)

where 119908 isin D

119870119899

119899+1(1198870 1198871 119887

119899)

= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)


3119887119899minus4

0(1198874+ 311988711198872)

+ sum119895ge5

119887119899minus119895

0119881119895

(4)


variables 1198871 1198872 119887

119899





119899|



119899| and




119899=1119901119899119911119899 be so that 119877119890(119901(119911)) gt 0


100381610038161003816100381610038161199012+ 1205721199012

1

10038161003816100381610038161003816le 2 + 120572

1003816100381610038161003816119901110038161003816100381610038162

(5)



119899=1119888119899119911119899


100381610038161003816100381610038161198882+ 1205741198882

1

10038161003816100381610038161003816le 1 + (120574 minus 1)

1003816100381610038161003816119888110038161003816100381610038162

(6)

Proof Write

119901 (119911) =[1 + 120593 (119911)]

[1 minus 120593 (119911)] (7)

where 119901(119911) = 1+suminfin



1= 21198881and 119901

2=

2(1198882+ 11988821)


2= 2(1198882+ 11988821) in (5) we

obtain100381610038161003816100381610038162 (1198882+ 1198882

1) + 120572 (2119888

1)210038161003816100381610038161003816le 2 + 120572

10038161003816100381610038162119888110038161003816100381610038162 (8)

or100381610038161003816100381610038161198882+ (1 + 2120572) 119888

2

1

10038161003816100381610038161003816le 1 + 2120572

1003816100381610038161003816119888110038161003816100381610038162

(9)




(119894)10038161003816100381610038161198870

1003816100381610038161003816 le

119861 minus 119860


119861 minus 119860 119900119905ℎ119890119903119908119894119904119890

(119894119894)10038161003816100381610038161198871

1003816100381610038161003816

le

119861 minus 119860


2 (119861 minus 119860)|1198870|2

119894119891 0 le 119860 le 1 minus 2119861

119861 minus 119860

2 119900119905ℎ119890119903119908119894119904119890

(119894119894119894)100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2

(119894V) 10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 119894119891 119887

119896= 0 119891119900119903 0 le 119896 le 119899 minus 1

(10)


1199111198921015840 (119911)

119892 (119911)= minus1 minus

infin

sum119899=0

119865119899+1

(1198870 1198871 1198872 119887

119899) 119911119899+1

(11)

where 119865119899+1

(1198870 1198871 1198872 119887




5=



119865119899+1

(1198870 1198871 119887

119899)


119899+1=119899+1

119860 (1198941 1198942 119894

119899+1) 1198871198941

01198871198942

1sdot sdot sdot 119887119894119899+1

119899

(12)

where

119860 (1198941 1198942 119894

119899+1)

= (minus1)(119899+1)+2119894

1+sdotsdotsdot+(119899+2)119894

119899+1

(1198941+ 1198942+ sdot sdot sdot + 119894

119899+1minus 1) (119899 + 1)

(1198941) (1198942) sdot sdot sdot (119894

119899+1)

(13)



119908ℎ1015840 (119908)


infin

sum119899=1

119865119899+1

(1198610 1198611 1198612 119861

119899) 119908119899+1

(14)

where119865119899+1

(1198610 1198611 1198612 119861


119899 + 1 given by

119865119899+1

= minus119899 (119899 minus (119899 + 1))

119899 (119899 minus 2119899)119861119899

0

minus119899 (119899 minus (119899 + 1))


01198611

minus119899 (119899 minus (119899 + 1))


01198612

minus119899 (119899 minus (119899 + 1))


0

sdot (1198613+119899 minus (2119899 minus 3)

21198612


119861119899minus119895

0119870119895

(15)


variables 1198611 1198612 119861

119899minus1and

1198610= minus1198870 119861

119899= minus

1

119899119870119899

119899+1(1198870 1198871 119887

119899) (16)


1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198881 1198882 119888

119899 119861) 119911119899

(17)

119908ℎ1015840 (119908)

ℎ (119908)= minus

1 + 119860120595 (119908)

1 + 119861120595 (119908)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198891 1198892 119889

119899 119861) 119908

119899

(18)


119899=1119888119899119911119899 and 120595(119908) = sum

infin

119899=1119889119899119908119899 are two



119899(1198961 1198962 119896

119899 119861) are given by

119870119901

119899(1198961 1198962 119896

119899 119861)

=119901

(119901 minus 119899)119899119896119899

1119861119899minus1

+119901


11198962119861119899minus2

+119901


11198963119861119899minus3

+119901


1

sdot [1198964119861119899minus4

+119901 minus 119899 + 3

21198962

2119861]

+119901


1

sdot [1198965119861119899minus5

+ (119901 minus 119899 + 4) 11989621198963119861]

+ sum119895ge6

119896119899minus119895

1119883119895

(19)


variables 1198962 1198963 119896

119899


minus119865119899+1

(1198870 1198871 1198872 119887

119899) = (119860 minus 119861)119870

minus1

119899+1(1198881 1198882 119888

119899+1 119861)

(20)


minus 119865119899+1

(1198610 1198611 1198612 119861

119899)

= (119860 minus 119861)119870minus1

119899+1(1198891 1198892 119889

119899+1 119861)

(21)


1198870= (119860 minus 119861) 119888

1

minus1198870= (119860 minus 119861) 119889

1

(22)

21198871minus 1198872

0= (119861 minus 119860) (119888

2

1119861 minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (119889

2

1119861 minus 1198892)

(23)




0and then adding

them gives

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus 1198611198882

1minus 1198611198892

1) (24)



21198872

0= (119861 minus 119860) (119888

2+ 1198892minus

2119861

(119861 minus 119860)21198872

0) (25)


sides we obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(119861 minus 119860)

2

(1198882+ 1198892)

2 (2119861 minus 119860)le

119861 minus 119860

radic2119861 minus 119860 (26)



119861 minus 119860



we obtain

41198871= (119861 minus 119860) [(119889

2minus 1198882) minus (119889

2

1minus 1198882

1) 119861] = (119861 minus 119860) (119889

2minus 1198882)

(28)


100381610038161003816100381611988711003816100381610038161003816 le

119861 minus 119860

2 (29)


21198871= (119861 minus 119860) (119889

2+ 1198601198892

1minus 2119861119889

2

1) (30)


100381610038161003816100381611988711003816100381610038161003816 le

1

2(119861 minus 119860) (

100381610038161003816100381610038161198892+ 1198601198892

1

10038161003816100381610038161003816+ 2119861

1003816100381610038161003816119889110038161003816100381610038162

)

le1

2(119861 minus 119860) (1 + (119860 minus 1)

1003816100381610038161003816119889110038161003816100381610038162

+ 211986110038161003816100381610038161198891

10038161003816100381610038162

)

=119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

(31)




2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

lt119861 minus 119860

2 (32)


21198871minus 1198872

0= (119861 minus 119860) (

119861

(119861 minus 119860)21198872

0minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (

119861

(119860 minus 119861)21198872

0minus 1198892)

(33)


100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2 (34)


(119899 + 1) 119887119899= (119860 minus 119861) 119888

119899+1 (35)


obtain

10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 (36)







0|2 given by






References



























1 Introductions


119891 (119911) = 119911 +

infin

sum120581=2

119886120581119911120581

(1)


119867[119886 119899] = 119901 isin 119867 119901 (119911) = 119886 + 119886119899119911119899

+119886119899+1

119911119899+1


(2)


119865 (119911) = 119911120573

+

infin

sum120581=2

120582120581119886120581119911120573+120581minus1

120573 isin N

120582120581= 120573120581minus1

119911 =1

120573 120573 |119911| lt 1

(3)


(119891 lowast 119892) (119911) =

infin

sum119896=0

119886119896119887119896119911119896

(4)


119896=0119886119896119911119896 and 119892(119911) = sum

infin

119896=0119887119896119911119896 are analytic in

119880

Let 120595(119911) = 119911120573 + sum

infin



120595 (119911) lowast119911120573

1 minus 120573119911 119911 =

1

120573 120573 isin N 120573 |119911| lt 1 (5)



119876(120485) (120572 120591 120574 120573)

= 119865 (119911) isin 119867 (120573) R119890

times120572[119865(119911)]

(120485)

[119911120573](120485)

+120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

gt 120574 119911 isin 119880

(6)

where

119911 = 0 120573 gt 120485 120485 isin N cup 0 also 120572 + 120591 = 120574 (7)







119891 (119911) =1

120572 + 120591int|119909|


times

infin

sum120581=0

(120572 + 120591) 119909120581

119911120581+1

(120581 + 1) 120591 + 120572] 119889120583 (119909)

(8)




Θ120572120573

(119911) = (120573int119911

0

119905120573minus120572minus1

[119891 (119905)]120572

119892 (119905) 119889119905)

1120573

(10)






119901 (119911) = int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909) (|119911| lt 1) (11)



119892 (119911) +1199111198921015840 (119911)

119888≺ ℎ (119911) (12)

then

119892 (119911) ≺ 119902 (119911) =119888119911minus119888120581

120581int119911

0

119905(119888minus120581)120581

ℎ (119905) 119889119905 ≺ ℎ (119911) (13)



1199111199011015840 (119911)

119901 (119911)≺ ℎ (119911) (14)

then

119901 (119911) ≺ 119902 (119911) = 119886 exp [1

119899int119911

0

ℎ (120585) 120585minus1

119889120585] (15)




(ii) |1199011015840(119911)| le 2R119890119901(119911)(1 minus 1199032)


100381610038161003816100381610038161003816100381610038161003816

1199111199011015840 (119911)

119901 (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 119901 (119911)

(1 minus 119903)2

(16)




(120485)

(120572120591120574120573)




119911120485

[119865 (119911)](120485)

=

120573

119875120485

(120572 + 120591)int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120573 minus 120485) (120572 + 120591) 119909120581119911120573+120581

(120572 + 120591) (120573 minus 120485) + 120581120591] 119889120583 (119909)

(17)

where 119899119875119903




119901 (119911)

=

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

isin 119867 [1 119899]

(18)

ByTheorem 3

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

= int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909)

(19)


[119865 (119911)](120485+1)

[119911120573](120485+1)

+120572[119865 (119911)]

(120485)

120591[119911120573](120485)

=1

120591int|119909|=1

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909119911

1 minus 119909119911119889120583 (119909)

(20)

which yields

119911((120572+120591)(120485minus120573)+120591)120591

int119911

0

[[119865 (120585)]

(120485+1)

[120585120573](120485+1)

+120572[119865 (120585)]

(120485)

120591[120585120573](120485)

]


119889120585

=1

120591int|119909|=1

119911((120572+120591)(120485minus120573)+120591)120591

times int119911

0

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909120585

1 minus 119909120585


119889120585 119889120583 (119909)

(21)



shows that 119865(119911) isin 119876(120485)(120572 120591 120574 120573)



119865 (119911) =1

120572 + 120591int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581119911120573+120581

120572120573 + 120591 (120581 + 120573)] 119889120583 (119909)

(22)


1




10038161003816100381610038161198861205811003816100381610038161003816 le 119867 (120572 120591 120574 120573) (23)

where

119867(120572 120591 120574 120573) =2 (120572 + 120591 minus 120574) (120573 minus 120485) [

120573

119875120485]

120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(24)


119911120485

[119865119909(119911)](120485)

120573119875120485

= 119911120573

+

infin

sum120581=2


(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)(|119909| = 1)

(25)




119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [[120581 minus 1] 119903

120581minus1]

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

le2119903R119890 Ψ (119911)

(1 minus 119903)2

(26)


119866 (119911) equiv120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

+120572[119865(119911)]

(120485)

[119911120573](120485)

minus 120574 (27)

and then119911Ψ1015840 (119911)

Ψ (119911)=

suminfin

120581=2120599 (120572 120591 120574 120573) [120581 minus 1] 119886

120581119911120581minus1

1 + suminfin

120581=2120599 (120572 120591 120574 120573) 119886

120581119911120581minus1

(28)

where

120599 (120572 120591 120574 120573) =120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591 minus 120574) [120573

119875(120485)

]

(29)

From (23) and (28) we got100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [120581 minus 1] 119903

120581minus1

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

(120581 ge 2) (30)


119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 Ψ (119911)

(1 minus 119903)2

(31)

Sinceinfin

sum120581=2


lt 2119903R119890 Ψ (119911)

1 +

infin

sum120581=2

2 (120573 minus 120485) 119903120581minus1

gt (1 minus 119903)2

(32)




(120485)

(120572 120591 120574 120573)



1199111198661015840 (119911)

119866 (119911) ≺ ℎ (119911) (33)

then

119866 (119911) ≺ 119902 (119911) = (120572 + 120591 minus 120574) exp [119899minus1

int119911

0

ℎ (120585) 120585minus1

119889120585 ] (34)



119866 (119911) = (120572 + 120591 minus 120574) +

infin

sum120581=2

120582120581[ [120573+120581minus1]119875

120485+1] 119886120581119911120581minus1

120573119875120485+1

(35)


1199111198661015840

(119911)

119866 (119911) ≺ ℎ (119911) (36)




1199111198661015840 (119911)

119866 (119911) ≺

radic119911

1 minus 119911(37)

then

119866 (119911) ≺ (120572 + 120591 minus 120574) 1 + radic119911

1 minus radic119911

1119899

120572 + 120591 = 120574 (38)




120573 gt 120485 + 1 with 120572 + 120591 = 0 If

119866 (119911) ≺ (120572 + 120591)1 + 119860119911

1 + 119861119911 (39)

then[119865(119911)]

(120485)

[119911120573](120485)

≺(120572 + 120591) (120573 minus 120485)

120591120581119911(120572+120591)(120485minus120573)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(40)


1

(120572 + 120591)119866 (119911) = 1 +

infin

sum120581=2

119862120581119886120581119911120581minus1

(41)

where

119862120581=

120582120581[ (120573+120581minus1)119875

120485+1] sdot [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591) [120573

119875120485+1

] (42)

Let 119892(119911) = [119865(119911)](120485)

[119911120573](120485) then 119892(119911) = 1 + sum

infin

120581=2120603120581119886120581119911120581minus1

where 120603120581= 120582120581[(120573+120581minus1)

119875120485+1

][(120573)

119875120485] and from (39) we have

119892 (119911) +120591

(120572 + 120591) (120573 minus 120485)1199111198921015840

(119911) ≺ (1 + 119860119911

1 + 119861119911) = ℎ (119911) (43)


119892 (119911) ≺(120572 + 120591) (120573 minus 120485)

120591120581119911minus(120572+120591)(120573minus120485)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(44)


Corollary 16 Let

119876(0)

(120572 120591 0 120573) =120572 (119865 (119911))

119911120573+

120591[119865 (119911)]1015840

[119911120573]1015840

(45)

If

119876(0)

(120572 120591 0 120573) ≺ (120572 + 120591) (1 + 119860119911

1 + 119861119911) (46)

then

119865 (119911)

119911120573≺

120572120573 + 120591 (120573 minus 120581)

120591120581119911minus120573(120572+120591)120591120581

times int119911

0

120585(120573(120572+120591)minus120591120581)120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(47)






Acknowledgments



References





















1 Introduction


119891 (119911) =119892 (119911)

ℎ (119911) (1)






R ℎ (119911) gt 0 (119911 isin 119880lowast

) (2)


119891 ≺ 119892 in 119880 or 119891 (119911) ≺ 119892 (119911) (119911 isin 119880lowast

) (3)


|119908 (119911)| le |119911| 119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880lowast

) (4)


119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0)

119891 (119880) sube 119892 (119880) (119911 isin 119880lowast

) (5)





2 Preliminaries


119891 (119911) =1

119911+

infin

sum119899=1

119886119899119911119899

(6)




119891119895(119911) =

1

119911+

infin

sum119899=1

119886119899119895119911119899

(7)


and 1198912(119911) by

(1198911lowast 1198912) =

1

119911+

infin

sum119899=1

11988611989911198861198992119911119899

(8)


119902120582120583

(119911) =1

119911+

infin

sum119899=1

(120582

119899 + 1 + 120582)

120583

119911119899

(120582 gt 0 120583 ge 0) (9)



119871120582120583119891 (119911) = (119891 (119911) lowast 119902

120582120583(119911))

=1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(10)


119905119886119892 and 119871


tively by

119871120582120583119892 (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198871198991003816100381610038161003816 119911119899


119871120582120583ℎ (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198881198991003816100381610038161003816 119911119899


ℎ (119911) isin Σ)

(11)


119871120582120583

(119891 lowast 119892) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast)

119871120582120583

(119891 lowast ℎ) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198881198991003816100381610038161003816 119911119899


)

(12)






property

120588119871120582120583

(119891 lowast 119892) (119911)

119871120582120583

(119891 lowast ℎ) (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911 (119911 isin 119880

lowast

) (13)


and 119871(120582 120583)(119891 lowast ℎ)(119911) = 0


120601 (120572 120573 119911) =1

119911+

infin

sum119899=0

(120572)119899+1

(120573)119899+1

119911119899

(14)



120601 (120572 120573 119911) =1

119911 21198651(1 120572 120573 119911) (15)

where

21198651(119887 120572 120573 119911) =

infin

sum119899=0

(119887)119899(120572)119899

(120573)119899

119911119899

119899(16)


120582120583(119911)


119871 (120572 120573 120582 120583) 119891 (119911) = (119891 (119911) lowast 120601 (120572 120573 119911) lowast 119902120582120583

(119911))

=1

119911+

infin

sum119899=1

(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(17)




119911(119871 (120572 120573 120582 120583) 119891 (119911))1015840

= 120572119871 (120572 + 1 120573 120582 120583) 119891 (119911)

minus (120572 + 1) 119871 (120572 120573 120582 120583) 119891 (119911) (18)


120588119871 (120572 + 1 120573 120582 120583) 119891 (119911)

119871 (120572 120573 120582 120583) 119891 (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911

(0 le 119861 lt 119860 le 1 120588 gt 0)

(19)



relation (19) above






infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861)

minus10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(20)


119891119899(119911) =

1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

(21)



1003816100381610038161003816100381610038161003816100381610038161003816

120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899) 119911119899+1

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899119911119899+1

)

minus11003816100381610038161003816100381610038161003816100381610038161003816

le 120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899)10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899

10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816)

minus1

le 1

(22)





100381610038161003816100381610038161003816100381610038161003816lt 1

(119911 isin 119880lowast)

(23)


Σ120583




120582(120588 119860 119861) then

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119899 ge 1

(24)


(21)


120582(120588 119860 119861)



1

119903minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903

le1003816100381610038161003816119891 (119911)

1003816100381610038161003816

le1

119903+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903


1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

le100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(25)



inequalityinfin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(26)


1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

le1

119903+ 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

119903+ 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|minus

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

ge1

119903minus 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

ge1

119903minus 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(27)


sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(119860 minus 119861) (2 + 120582)

120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(28)

Hence

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2+

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

le1

1199032+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

ge1

1199032minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816

ge1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(29)



120582(120588 119860 119861) which




disk |119911| lt 1199031 where

1199031= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(30)



119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (31)

For |119911| lt 1199031 we have

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 1 + 120582))

120583

119886119899119911119899

1119911 + suminfin

119899=1(120582(119899 + 1 + 120582))

120583

119886119899119911119899

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

100381610038161003816100381610038161003816100381610038161003816

lesuminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(32)


sum119899=1

(119899 + 1) (120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1


infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

)

(33)

or

suminfin

119899=1(119899 + 2 minus 120575) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(1 minus 120575)le 1 (34)


(119899 + 2 minus 120575) (120582(119899 + 1 + 120582))120583

|119911|119899+1

(1 minus 120575)

le (120582

119899 + 120582 + 1)

120583

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899 ge 1

(35)



|119911| le (1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899+1

(36)




disk |119911| lt 1199032 where

1199032= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(37)



1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891 (119911))10158401015840

(119891 (119911))1015840+ 2

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (38)






infin

sum119899=1

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(39)


119891119899(119911)

=1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

119899 ge 1

(40)




120582(120588 120572 120573 119860 119861) then1003816100381610038161003816119886119899

1003816100381610038161003816

le(119860 minus 119861) (119899 + 120582 + 1)

120583

(120573)119899+1

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861)) (120572)

119899+1

119899 ge 1

(41)


(40)




1199033= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(42)





1199034= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(43)






Acknowledgment


References






















1 Introduction


Φ (119911 119904 119886) =

infin

sum119899=0

119911119899

(119899 + 119886)119904


0 119904 isin C when |119911| lt 1

R (119904) gt 1 when |119911| = 1)

(1)


119904(120585) defined by

120577 (119904) =

infin

sum119899=1

1

119899119904= Φ (1 119904 1) = 120577 (119904 1) (R (119904) gt 1)

120577 (119904 119886) =

infin

sum119899=0

1

(119899 + 119886)119904


0)

ℓ119904(120585) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 1)119904= Φ (e2120587119894120585 119904 1) (R (119904) gt 1 120585 isin R)

(2)


119904(119911)

Li119904(119911) =

infin

sum119899=1

119911119899

119899119904= 119911Φ (119911 119904 1)


(3)


120601 (120585 119886 119904) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 119886)119904= Φ (e2120587119894120585 119904 119886)



when 120585 isin Z)

(4)




Φ (119911 119904 119886) =1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

1 minus 119911eminus119905d119905

(R (119886) gt 0R (119904) gt 0 when |119911| ≦ 1 (119911 = 1) R (119904) gt 1

when 119911 = 1) (5)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) =

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

119911119899

(119899 + 119886)119904

(119901 119902 isin N0 120582

119895isin C (119895 = 1 119901)

119886 120583119895isin C Z

minus

0(119895 = 1 119902)

120588119895 120590

119896isin R

+

(119895 = 1 119901 119896 = 1 119902)





lowast

)

(6)

with

nablalowast

= (

119901

prod119895=1

120588minus120588119895

119895) sdot (

119902

prod119895=1

120590120590119895

119895)

Δ =

119902

sum119895=1

120590119895minus

119901

sum119895=1

120588119895

Ξ = 119904 +

119902

sum119895=1

120583119895minus

119901

sum119895=1

120582119895+119901 minus 119902

2

(7)



(120582)120581=Γ (120582 + 120581)

Γ (120582)

=


1 (120581 = 0 120582 isin C 0)

(8)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0)

(9)



119898or

119897Ψ119898(119897 119898 isin N

0) involved


119897119865119898(119897 119898 isin N




1198871 119887

119898such that

119886119895isin C (119895 = 1 119897) 119887

119895isin C Z

minus

0(119895 = 1 119898)

(10)


119897Ψlowast

119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

=

infin

sum119899=0

(1198861)1198601119899sdot sdot sdot (119886

119897)119860119897119899

(1198871)1198611119899sdot sdot sdot (119887

119898)119861119898119899

119911119899

119899

=Γ (119887


119898)

Γ (1198861) sdot sdot sdot Γ (119886

119897) 119897Ψ119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

(119860119895gt 0 (119895 = 1 119897) 119861

119895gt 0 (119895 = 1 119898)

1 +

119898

sum119895=1

119861119895minus

119897

sum119895=1

119860119895≧ 0)

(11)


|119911| lt nabla = (

119897

prod119895=1

119860minus119860119895

119895) sdot (

119898

prod119895=1

119861119861119895

119895) (12)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

Γ (119904)intinfin

0


119905120582)

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0)

(13)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 0 120582)

= Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

= e119887Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 0)

(14)


119901 minus 1 = 119902 = 0 (1205821= 120583 120588

1= 1) (15)


Θ120582

120583(119911 119904 119886 119887)

=1

Γ (119904)intinfin

0


119905120582) (1 minus 119911eminus119905)

minus120583

d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0 120583 isin C)

(16)


R (119904) gt 0 when 119887 = 0 |119911| ≦ 1 (119911 = 1) (17)

or

R (119904 minus 120583) gt 0 when 119887 = 0 119911 = 1 (18)




Θ120582


Θ120582

120583(119911 119904 119886 0) = Φ

lowast

120583(119911 119904 119886) =

1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

(1 minus 119911eminus119905)120583d119905

(R (119886) gt 0 R (119904) gt 0 when |119911| ≦ 1 (119911 = 1)

R (119904 minus 120583) gt 0 when 119911 = 1)

(19)



[4] as follows

Φlowast

120583(119911 119904 119886) =

infin

sum119899=0

(120583)119899

(119886 + 119899)119904

119911119899

119899 (20)


120583(119911 119904 119886 119887) that is


Θ1

1(1 119904 119886 119887)

= 120577119887(119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1


119905) d119905

(21)




(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899

119899(120582 gt 0)

(22)



119867119898119899

pq (119911)

= 119867119898119899

pq [119911 |(119886

1 119860

1) (119886p 119860p)

(1198871 119861

1) (119887q 119861q)

]

=1

2120587119894intL

Ξ (119904) 119911minus119904d119904 (119911 isin C 0

1003816100381610038161003816arg (119911)1003816100381610038161003816 lt 120587)

(23)

where

Ξ (119904) =prod

119898

119895=1Γ (119887

119895+ 119861

119895119904) sdot prod

119899

119895=1Γ (1 minus 119886

119895minus 119860

119895119904)

prodp

119895=119899+1Γ (119886

119895+ 119860

119895119904) sdot prod

q

119895=119898+1Γ (1 minus 119887

119895minus 119861

119895119904)

(24)


119895gt 0 (119895 =

1 p) 119861119895gt 0 (119895 = 1 q) 119886

119895isin C (119895 = 1 p)



Γ (119887119895+ 119861

119895119904)

119898

119895=1

(25)


Γ (1 minus 119886119895+ 119860

119895119904)

119899

119895=1

(26)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902





1 minus 119905)

=

infin

sum119899=0

119871(120572)

119899(119909) 119905

119899

(|119905| lt 1 120572 isin C)

(27)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=eminus119887

Γ (119904)

infin

sum119899=0

119899

sum119896=0

(minus1)119896

(119899

119896) sdot Γ (119904 + 120582 (120572 + 119896 + 1))

sdot 119871(120572)

119899(119887)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 + 120582 (120572 + 119895 + 1) 119886)

(R (119886) gt 0 R (119904 + 120582120572) gt minus120582)

(28)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) (29)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



D120572

119911119911

119901

119891 (119911) =1

Γ (minus120572)int119911

0

119891 (120585) 120585119901


(R (120572) lt 0 R (119901) gt 1)

(30)


D120572

119911119911

119901

119891 (119911) =d119898

d119911119898D

120572minus119898

119911119911

119901

119891 (119911) (31)



Im(120585)

Branch line for

Branch line for


exp[p(ln(g(120585)))]

C1

C2

C3

C4

Re(120585)

z

a

gminus1(0)





119901


119863120572

119892(119911)[119892 (119911)]

119901

119891 (119911)

=eminus119894120587119901Γ (1 + 120572)4120587 sin (120587119901)

times int119862(119911+119892

minus1(0)+119911minus119892

minus1(0)minus119865(119886)119865(119886))

sdot119891 (120585) [119892 (120585)]

119901

1198921015840 (120585)

[119892 (120585) minus 119892 (119911)]120572+1

d120585

(32)



1cup119862

2cup119862

3cup119862








119892(119911)[119892(119911)]

119901



119863120572

119911119911

119901

=Γ (1 + 119901)

Γ (1 + 119901 minus 120572)119911119901minus120572

(R (119901) gt minus1) (33)



11205881199011205901120590119902)

12058211205821199011205831120583119902


119863120572

119911119911

120573minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899)

1198871120582

| (119904 1) (01

120582)]119863

120572

119911119911

120573minus1119911119899

119899

=Γ (120573)

120582Γ (119904) Γ (120573 minus 120572)

infin

sum119899=0

(120573)119899

(120573 minus 120572)119899

sdot

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899+120573minus120572minus1

119899

=Γ (120573)


Φ(1205881120588119901112059011205901199021)

12058211205821199011205731205831120583119902120573minus120572

(119911 119904 119886 119887 120582)







119863120572

119892(119911)119891 (119911) = 119863

120572

ℎ(119911)119891 (119911) 119892

1015840(119911)

ℎ1015840(119911)(ℎ(119908) minus ℎ(119911)

119892(119908) minus 119892(119911))

120572+1

100381610038161003816100381610038161003816100381610038161003816119908=119911

(35)




119911119901119874120572

120573119891 (119911)

= 119901(119911119901minus1

)minus120572

sdot119911

119874120572

120573119891 (119911) (119911

119901minus1

)120572

119901minus1

prod119904=1

(1 minus119911

119908eminus(2120587119894119904)119901)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(36)

where

119892(119911)119874120572


Γ (120573)

Γ (120572)[119892 (119911)]

1minus120573

119863120572minus120573

119892(119911)[119892 (119911)]

120572minus1

sdot sdot sdot

(37)



120579 (119911) = (119911 minus 1199110) 119902 (119911) (38)


119870

=0 1 [119888] ([119888] 119905ℎ119890 119897119886119903119892119890119904119905 119894119899119905119890119892119890119903 119899119900119905 119892119903119890119886119905119890119903 119905ℎ119886119899 119888)

(39)


0and let

120596 = exp (2120587119894119886) (40)



sum119896isin119870

119888minus1

120596minus120574119896

119891 (120579minus1

(120579 (119911) 120596119896

))

=

infin


[120579 (119911)]119888119899+120574

Γ (119888119899 + 120574 + 1)

sdot119863119888119899+120574

119911minus119887119891 (119911) 120579

1015840

(119911) (119911minus119911

0

120579 (119911))

119888119899+120574+1

100381610038161003816100381610038161003816100381610038161003816119911=1199110

(1003816100381610038161003816119911minus1199110

1003816100381610038161003816=100381610038161003816100381611991101003816100381610038161003816)

(41)



119891 (119911) = 119888

infin


(119911 minus 1199110)119888119899+120574

Γ(119888119899 + 120574 + 1)119863

119888119899+120574

119911minus119887119891(119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199110

(42)



1)(119911minus119911

2)) where 119911

1and 119911

2are two



120596 = exp(2120587119894119886) (43)


1and 119911


curves

119862 (119905) 119862 (119905) subR 0 lt 119905 ≦ 119903 (44)

be defined by

119862 (119905) = 1198621(119905) cup 119862

2(119905)

= 119911 1003816100381610038161003816120582119905 (1199111 1199112 119911)

1003816100381610038161003816 =1003816100381610038161003816100381610038161003816120582119905(119911

1 119911

21199111+ 119911

2

2)1003816100381610038161003816100381610038161003816

(45)

where

120582119905(119911

1 119911

2 119911) = [119911 minus

1199111+ 119911

2

2+ 119905 (

1199111minus 119911

2

2)]

sdot [119911 minus (1199111+ 119911

2

2) minus 119905 (

1199111minus 119911

2

2)]

(46)


1+ 119911



1199111+ 119911

2

2+ (

1199111minus 119911

2

2) 119905 (47)


2

2minus (

1199111minus 119911

2

2) 119905 (48)

Im(120585)

Branch line for

Branch line for

C1

C2

z1

z2

Re(120585)

z

c =z1 + z2

2

(120585 minus z2)120583120579(120585)

aminus120574minus1

(120585 minus z1)120579(120585)

aminus120574minus1



[(119911 minus 1199111) (119911 minus 119911

2)]

120582


2))))

(49)


plusmn

defined by

119871plusmn=

119911 119911 =1199111+ 119911

2

2plusmn 119905 (

1199111minus 119911

2

2) (0 ≦ 119905 ≦ 1)

119911 119911 =1199111+ 119911

2

2plusmn 119894119905 (

1199111minus 119911

2

2) (119905 lt 0)

(50)



1)(119911minus119911

2) gt 0 (v)


2)119901


+cup 119871


119911minus1199112

(119911 minus 1199112)119901


119870 = 119896 119896 isin N arg(120582119905(119911

1 119911

21199111+ 119911

2

2))

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) +

2120587119896

119886

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) + 2120587

(51)


defined by

120585 =1199111+ 119911

2

2+1199111minus 119911

2


sum119896isin119870

119888minus1120596minus120574119896

1199111minus 119911

2

119891 (120601minus1

(120596119896

120601 (119911)))

times [120601minus1

(120596119896

120601(119911)) minus 1199111]][120601

minus1

(120596119896

120601(119911)) minus 1199112]120583

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

[120601 (119911)]119888119899+120574

(52)


where

120601 (119911) =119911 minus 119911

1

119911 minus 1199112

(53)



2)120583

1199111minus 119911

2

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

(119911 minus 119911

1

119911 minus 1199112

)

119888119899+120574

(54)


2) where 119911

1and 119911




sum119896isin119870

119888minus1

120596minus120574119896

[(1199112minus 119911

1+ radicΔ

119896

2)

120572

(1199111minus 119911

2+ radicΔ

119896

2)

120573

sdot 119891 (1199111+ 119911

2+ radicΔ

119896

2)


sdot (1199112minus 119911

1minus radicΔ

119896

2)

120572


2minus radicΔ

119896

2)

120573

119891(1199111+ 119911

2minus radicΔ

119896

2)]

=

infin



eminus119894120587119888(119899+1)[120579 (119911)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911minus1199112


(120579 (119911)

(119911minus1199112) (119911minus119911

1))


1205791015840

(119911)119891 (119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199111

(55)

where

Δ119896= (119911

1minus 119911

2)2

+ 4119881 (120596119896

120579 (119911))

119881 (119911) =

infin

sum119903=1

119863119903minus1

119911[119902 (119911)]

minus119903

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

119911119903

119903

120579 (119911) = (119911 minus 1199111) (119911 minus 119911

2) 119902 ((119911 minus 119911

1) (119911 minus 119911

2))

(56)


(119911 minus 1199111)(119911 minus 119911

2)) and 119911


119891 (119911) = 119888119911minus120573

(119911 minus 1199111)minus120572

times

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot119863minus120572+119888119899+120574

119911119911


(119911 + 119908 minus 1199111)119891(119911)

10038161003816100381610038161003816119911=1199111

(119908=119911)

(57)



119863120572

119911119911

119901+119902

119906 (119911) V (119911)

=

infin


(120572

120574 + 119899)119863

120572minus120574minus119899

119911119911

119901

119906 (119911)119863120574+119899

119911119911

119902V (119911) (58)




119863120572

119911119911

120572+120573minus1

119906 (119911) V (119911)


sdot

infin


119863120572+]+1minus119899119911



(60)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 (1120582))]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot 119865(119896minus1)



where 120582 gt 0 and 119865(119899)119863


119865(119899)

119863[119886 119887

1 119887

119899 119888 119909

1 119909

119899]

=

infin

sum1198981119898119899=0

(119886)1198981+sdotsdotsdot+119898

119899

(1198871)1198981

sdot sdot sdot (119887119899)119898119899

(119888)1198981+sdotsdotsdot+119898

119899

1199091198981

1


119909119898119899

119899

119898119899

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091198991003816100381610038161003816 lt 1)

(62)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119896

(119911119896minus1)120572

sdot119911119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) (119911119896minus1

)120572

sdot

119896minus1

prod119904=1

(1 minus119911

119908eminus2120587119894119904119896)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(63)




119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

(64)



119896

(119911119896minus1)120572

119911

119874120572

120573

times

1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119911119899+(119896minus1)120572

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


sdot((119911119908) eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119908=119911

=119896

120582Γ (119904) (119911119896minus1)120572

times

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981



sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

times119911

119874120572

120573119911


119896minus1+119899+(119896minus1)120572

=119896

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


times(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

sdotΓ (120573) Γ (119898


119896minus1+ 119899 + 119896120572)

Γ (120572) Γ (120573 + 1198981+ sdot sdot sdot + 119898

119896minus1+ 119899 + 119896120572)


=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 1120582)]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(119899+119896120572)1198981+sdotsdotsdot+119898

119896minus1

(1+120572minus120573)1198981



119896minus1

sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

(65)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


119911minus1198881198990(119911 minus 119911

0)119888119899

Γ (119888119899 + 1) Γ (1 minus 119888119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(1003816100381610038161003816119911 minus 1199110

1003816100381610038161003816 =100381610038161003816100381611991101003816100381610038161003816 120582 gt 0)

(66)


Proof Setting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


0 we

have

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


(119911 minus 1199110)119888119899

Γ (1 + 119888119899)

sdot119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

(67)


0| = |119911

0|


119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

=119911minus1198881198990

Γ (1 minus 119888119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(68)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(69)


119911 =1199111

2+1199111



Proof By taking 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 9 with 119911

2= 0 120583 = 120572 ] = 120573 and 0 lt 119888 ≦ 1 we find

that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888(119911 minus 1199111)minus120573

119911minus120572

1199111

sdot

infin



sdot (119911 minus 119911

1

119911)119888119899+120574

sdot119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

(71)



119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120572+120573minus1

1

Γ (120572 + 119888119899 + 120574)

Γ (120572 + 120573)

times Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582)

(72)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573+120574

(119911 minus 1199111)minus120572+120574

119911120573+120572minus2120574minus1

1

sdot

infin


sin [(120573 minus 119888119899 minus 120574) 120587] e119894120587119888(119899+1)

sin [(120573 + 119888 minus 120574) 120587] Γ (1 minus 120572 + 119888119899 + 120574)

sdot (119911(119911 minus 119911

1)

11991121

)

119888119899

Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdot [(119911 minus 1199111)Φ

(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (120573 minus 119888119899 minus 120574

120572 + 120573 minus 2119888119899 minus 2120574) 119911

1

sdotΦ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)]

(73)


119911 =1199111

2+1199111



Proof Putting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 10 with 119911

2= 0 0 lt 119888 ≦ 1 119902(119911) = 1 and 120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2) we find that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573

(119911 minus 1199111)minus120572

sdot

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

(75)


119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

= 119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

+ (119911 minus 1199111)119863

minus120572+119888119899+120574

119911


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120573+120572minus2119888119899minus2120574

1


Γ (1 + 120573 + 120572 minus 2119888119899 minus 2120574)

sdot Φ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (119911 minus 119911

1

1199111

)Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdotΦ(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582) )

(76)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

=Γ (120591) Γ (1 + ] minus 120591) sin (120574120587)

120587

sdot

infin


(minus1)119899

Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)


(77)



12058211205821199011205831120583119902


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=

infin


(] minus 120591120574 + 119899


119911]minus1

sdot 119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(78)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)



119911]minus1 =

Γ (])Γ (120591 + 120574 + 119899)

119911120591+120574+119899minus1

119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911minus120574minus119899

Γ (1 minus 120574 minus 119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(79)



Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

= (Γ (120591) Γ (1 + ] minus 120591) sin (120573120587)




sdotsin (120579120587)

sin [(120579 + 120574) 120587]

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(80)



120572 = ] minus 120591 119906 (119911) = 119911120591minus120573

V (119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(81)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


sdot

infin


119863]minus120591+120574+1minus119899119911




119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(82)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

119863]minus120591+120574+1minus119899119911


=Γ (] minus 120579 minus 119899)


119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (120579 + 119899)

Γ (1 + 120579 + 120574)

times 119911120579+120574

Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(83)





lim119887rarr0

11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)] = 120582Γ (119904)

(120582 gt 0)

(84)



Φ(120588112058811990113120590

112059011990213)

12058211205821199011205721205831120583119902120573

(119911 119904 119886)

=3Γ (120573) Γ (3120572)

Γ (120572) Γ (120573 + 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

(3120572)119899

(120573 + 120572)119899

119911119899

(119899 + 119886)119904


(85)


1198651[119886 119887

1 119887

2 119888 119909

1 119909

2]

=

infin

sum11989811198982=0

(119886)1198981+1198982

(1198871)1198981

(1198872)1198982

(119888)1198981+1198982

1199091198981

1

1198981

1199091198982

2

1198982

(max 100381610038161003816100381611990911003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 lt 1)

(86)




1= 120583 in



Θ120582

120583(119911 119904 119886 119887)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(111)

120583120572+119888119899+120574120572+120573(119911

1 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(87)


119911 =1199111

2+1199111





Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886)

=Γ (120591) Γ (1 + ] minus 120591)



times (Γ (1 + 120574 + 120579) sin [(] minus 120591 + 120573) 120587]


sdot

infin


Γ (]minus120579minus119899) Γ (120579+119899)Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120582

(119911 119904 119886)





References












































Contents





















Toka Diagana










1 Introduction




119886(119899 119896) and 119878



(119905)119899=

119899

sum119896=0

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

(119905 minus 119886)119896

(1)

(119905 minus 119886)119899

=

119899

sum119896=0

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(119905)119896 (2)


119904119886(119899 119896) =

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

119878119886(119899 119896) =

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(3)


119904119886(119899 + 1 119896) = 119904

119886(119899 119896 minus 1) + (119886 minus 119899) 119904

119886(119899 119896) (4)

119878119886(119899 + 1 119896) = 119878

119886(119899 119896 minus 1) + (119896 minus 119886) 119878

119886(119899 119896) (5)


119904119886(0 0) = 1 119904

119886(119899 0) = (119886)

119899 119904

119886(0 119896) = 0 119899 119896 = 0

119878119886(0 0) = 1 119878

119886(119899 0) = (minus119886)

119899

119878119886(0 119896) = 0 119899 119896 = 0

(6)


1+ 119883


119896of





119875 [119882 = 119899] =119896

(1 minus 120579)minus119904

(minus log (1 minus 120579))119896

120579119899

119899(minus1)

119899minus119896

119904minus119904(119899 119896)

119875 [119885 = 119899] =119896

119890119898 (119890119897 minus 1)119896

119897119899

119899(minus1)

119899minus119896

119878minus119898119897

(119899 119896)

(7)






119870(1)

119899119903= 119903 + [log(119899 minus 1


1

119903+ 119900 (1)] (8)


2 Explicit Formula



119865119899(119909) = prod

119899


1198651015840

119899(119909) = sum


119899minus1le119899

119894isinN11989911989511198952119895119899minus1

1198911015840

119894(119909)

119899minus1

prod119896=1

119891119895119896(119909) (9)


1 2

119889

119889120585(120585119895 + 119886)

1= 119895

119889

119889120585(120585119895 + 119886)

2= (120585119895 + 119886) 119895 + (120585119895 + 119886 minus 1) 119895

(10)


1

119895119899119889

119889120585(120585119895 + 119886)



119899minus1

prod119896=1

(120585 +119886 minus 119895

119896

119895) (11)



1

119895119899119889119896

119889120585119896(120585119895 + 119886)


119899minus119896le119899minus1

119896

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(12)


1

119895119899119889119898

119889120585119898(120585119895 + 119886)


119899minus119898le119899minus1

119898

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(13)

Then

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899


119899minus119898le119899minus1

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(14)




119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895) = sum



prod119902=1

(120585 +119886 minus 119894

119902

119895)

(15)

119894119902

isin 1198951 1198952 119895

119899minus119898 has ( 119899minus119898





119902=1(120585 + (119886 minus





119895119899minus119898minus1





119902=1(120585 + (119886 minus 119895


of (1119895119899)(119889119898+1119889120585119898+1)(120585119895 + 119886)119899 Thus we have

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899



119898 + 1

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)



119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(16)

Lemma 2 Let 119904(119899 119896 119886) =

(1119896)lim120585rarr0

((sum119896

119895=0(minus1)

119896minus119895

( 119896119895) (119889119896119889120585119896)(120585119895 + 119886)

119899)119896)

Then


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (17)

Proof Using Lemma 1

119904 (119899 119896 119886)

=1

119896lim120585rarr0

((

119896

sum119895=0


(119896

119895) 119895119899


119899minus119896le119899minus1

119896

times

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)) (119896)

minus1

)

(18)



(120585 + (119886 minus 119895119902)119895) = (1119895119899minus119896)prod

119899minus119896

119902=1(120585119895 + 119886 minus 119895

119902)


1

119896lim120585rarr0

[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(120585119895 + 119886 minus 119895119902)]

]

=1

119896[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)]

]

(19)

which boils down to


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (20)

since

1

119896

119896

sum119895=0


(119896

119895) 119895119896

= 119878 (119896 119896) = 1 (21)



119904119886(119899 119896) = 119904 (119899 119896 119886) = sum


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (22)

Proof We know that


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (23)



119899+1minus119896)







119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (24)

is equal to


119899minus119896+1le119899minus1

119899minus119896+1

prod119902=1

(119886 minus 119895119902) + (119886 minus 119899)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)

(25)

This implies that

119904 (119899 + 1 119896 119886) = 119904 (119899 119896 minus 1 119886) + (119886 minus 119899) 119904 (119899 119896 119886) (26)



119886(119899 119896)


120582 = (1205821ge 120582


119896) (27)


119894


14= 11989115

= 11989123

= 11989131

= 11989142

= 1 119891119894119895= 0 elsewhere

(1 le 119895 le 120582119894)

0 0 0 1 10 0 11 0 00 10

(28)



(119886119894)119894ge0



1198860 1198861 119886

ℎ then |119879119889119860(ℎ 119903)| = ( ℎ+1




120596-tableau is











119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)


120596 (|119888|) (29)




is equal to 119904119886(119899 119896)



0 1 2 119899 minus 1 and

120596119860(120601) = prod

119888isin120601

120596 (|119888|)

=

119899minus119896

prod119894=1

(119886 minus 119895119894) 119895

119894isin 0 1 2 119899 minus 1

(30)

If 119886 = 1198861+ 1198862for some 119886

1and 119886

2 then with 120596lowast(119895) = 119886

2minus 119895

120596119860(120601) =

119899minus119896

prod119894=1

(1198861+ 120596lowast

(119895119894))

=

119899minus119896

sum119903=0

119886119899minus119896minus119903

1sum


119903le119895119899minus119896

119903

prod119894=1

120596lowast

(119902119894)

(31)






may write

120596119860(120601) = sum

120595isin119861120601

120596119860(120595) (32)


120596119860(120595) = 119886

119899minus119896minus119903

1

119903

prod119894=1

120596lowast

(119902119894) (33)

where 1199021 1199022 119902

119903isin 119895

1 1198952 119895


written as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

sum120595isin119861120601

120596119860(120595) (34)



1 1198952 119895

119899minus119896



(119899

119896)(

119899 minus 119896

119903) (35)





( 119899119896) ( 119899minus119896

119903)

( 119899119903)

= (119899 minus 119903

119896) (36)


119904119886(119899 119896) =

119899minus119896

sum119903=0

(119899 minus 119903

119896) 119886119899minus119896minus119903

1sum120595isin119861119903


120596lowast

(|119888|) (37)



119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

1sum



120596lowast

(|119888|) (38)

Clearly 119861119899minus119895



tity

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

11199041198862

(119899 119895) (39)


1and 119886

2





119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (40)






2isin 1198791198891198602(119895 minus 1 119895 minus 119899+ 119896)


2 we obtain an


119860

(119898 +


sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601)

=

119899

sum119896=0

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011)

times

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

(41)


Note that

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)


119895minus119899+119896le119898+119895minus1

119895minus119899+119896

prod119902=1

(119886 minus 119892119902)



119895minus119899+119896

prod119902=1

(119886 minus (119898 + 119892119902))

= 119904119886minus119898

(119895 119899 minus 119896)

(42)


sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011) = 119904

119886(119898 119896)

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601) = 119904

119886(119898 + 119895 119899)

(43)

Thus

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (44)



convolution formula

119904119886(119899 + 1119898 + 119895 + 1) =

119899

sum119896=0

119904119886(119896 119899) 119904

119886minus(119896+1)(119899 minus 119896 119895) (45)

Proof Let


in 1198601= 0 1 119896 minus 1


are in 1198602= 119896 + 1 119899








21199092 + sdot sdot sdot + 119886

119899119909119899 has only

real zeros then

1198862

119896ge 119886119896+1

119886119896minus1

119896

119896 minus 1

119899 minus 119896 + 1

119899 minus 119896(119896 = 2 119899 minus 1) (46)


sum119896=0

119904119886(119899 119896) (119905 + 119886)

119896

(47)



119896=0119904119886(119899 119896)119905119896 are 119886 119886minus1 119886minus



119899


and hence unimodal


⟨119905⟩119899=

119899

sum119896=0


119904119886(119899 119896) (119905 + 119886)

119896

(48)



119904119886(119899 119896) = (minus1)

119899minus119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902) (49)


119886(119899 119896)| as

1003816100381610038161003816119904119886 (119899 119896)1003816100381610038161003816 = (minus1)

119899minus119896

119904119886(119899 119896)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902)

(50)



lt 1199062


infin

sum119894=1

1

119906119894

= infin

infin

sum119894=1

1

1199062119894

lt infin (51)




which sum119899119896


119870119899= 119899 minus [

119899

sum119894=1

1

119906119894

minus

119899

sum119894=1

1

1199062119894

(1 +1

119906119894

)

minus1

+ 119900 (1)] (52)


1

1+1

2+ sdot sdot sdot +

1

119899= log 119899 + 120574 + 119900 (1) (53)



119886(119899 119896)|

119899

119896=0



119899


mation

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (54)




Denoting by 119896119886119899



1= 119887 + 0 119906

2= 119887 + 1 119906


119896119886119899

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894)2(1 +

1

119887 + 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894) (119887 + 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894 + 1+ 119900 (1)]

(55)


119899minus1

sum119894=0

1

119887 + 119894 + 1= log (119887 + 119899) minus log 119887 (56)

From this we get

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (57)



119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (58)



1)(119886 minus


119899minus119896)where 119895


1 By Theorem 10 119904119886(119899 119896) = sum

119899119899minus119896 Thus with 119906

1= 119886 119906

2=

119886 minus 1 119906119899minus1

= 119886 minus 119899 + 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894)2(1 +

1

119886 minus 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894) (119886 minus 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119886 minus 119894 + 1+ 119900 (1)]

(59)


119899119896 0 1 2 30 11 12 2 13 6 6 14 24 35 10 15 120 225 85 156 720 1624 735 1757 5040 13132 6769 19608 40320 109584 118124 672849 362880 1026576 1172700 72368010 3628800 10628640 12753576 8409500


119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (60)


9


119896minus19

= [log(1 + 8

1) + 119900 (1)]

= [log 9 + 119900 (1)]

asymp 2

(61)


10


1198961010

= [log( 10 + 1

10 minus 10 + 1) + 119900 (1)]

= [log 11 + 119900 (1)]

asymp 2

(62)







(63)


Table 2 Values of 11990410(119899 119896)

119899119896 0 1 2 30 11 10 12 90 19 13 720 242 27 14 5040 2414 431 345 30240 19524 5000 6356 151200 127860 44524 81757 604800 662640 305956 772248 1814400 2592720 1580508 5376289 3628800 6999840 5753736 265576410 3628800 10628640 12753576 8409500


119904(119899 119896) 0 1 2 3 4 50 11 0 12 0 1 13 0 2 3 14 0 6 11 6 15 0 24 50 35 10 16 0 120 274 225 85 157 0 720 1764 1624 735 1758 0 540 13068 13132 6769 19609 0 40320 109584 118124 67284 2244910 0 362880 1026576 1172700 723680 269325





2= 2 119906


119896119899= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

1

1198942(1 +

1

119894)minus1

+ 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

(1

119894minus

1

119894 + 1) + 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894 + 1+ 119900 (1)]

(64)


119899minus1

sum119894=1

1

1 + 119894= log 119899 minus log 1 + 120574 + 119900 (1) (65)

Therefore we have

119896119899= [log 119899 + 120574 + 119900 (1)] (66)



7

119896=0occurs at

1198967= [log 7 + 120574 + 119900 (1)]

= [195 + 05772 + 119900 (1)]

= [253 + 119900 (1)]

asymp 2

(67)


11989610

= [log 10 + 120574 + 119900 (1)]

= [230 + 05772 + 119900 (1)]

= [28772 + 119900 (1)]

asymp 3

(68)




Acknowledgment


References
















1 Introduction















2 New Instructions







x

f(x)

i

(ii) and (iii)1

c













2119888 and



















2minus119890


minus119873

2minus119873

] (1)









2119909 multiplica-








1sdot finite

1+ finite

2= 2 (2)








2120583 in the case






2((maximum



Inputsign exp mant

Drop leading zeros

numZeros + 1


Add

Switch

s e12 m51

lookup output


exp

Add 1

expP1

adjustedE

Switch

bitRot


Add 0x7ff

Add 0x3ff

Switch

Concatenate bit

xor

Add drop carry

moreThan22Zeros

isNot12

Concatenate bit

51

42

42 + 1

44

30

31

52

52

52

52


Add



(c)

(b)

(d)

(e)

(f)

(g)


(h)

(i)

(j)

(k)

leading12Mant

(a)

(p)

(r)

(o)

(l)

(m)

(n)

(q)

(s)

(t)

(u)

(v)

5

sim

sim

0x7ffSubs

Subs 0b0 0




12-bit lookup

12-bit lookup

ge1

e(last 10bits)lt1

sim


0b10 0Subs

Subs 0b0 0

low 10bits4

6

11

11

11

11

12

Lower 10






2119872

recip 3 113 376 0500 2048 3 infin






3 Evaluation





4 Conclusion




(a)


recip (2minus119890

119888)ext (2minus119890



119888)sat

sqrt (2minus119890

119888)ext 21198902

119888 0infin 0 NaN 0 0 0 NaN




(b)


minusinfin 119888 119888 0 0

infin 119888 0 119888 0

NaN 119888 119888 119888 0

(c)



minusinfin119891

minusinfin119891

infin infin119891

infin119891

infin119891

infin119891

NaN0

NaN119891

NaN119891

NaN119891

NaN119891

NaN1

2119891

2119891

2119891

2119891

NaN2

1radic2119891

1radic2119891

1radic2119891

1radic2119891

NaN3

0119891

0119891

0119891

0119891

recipdiv

sqrtrsqrtcbrt

rcbrtqdrt

rqdrtlog2

loglog1pexp2

expexpm1

atan2atanasinacos

sincostan

0 5 10 15 20 25 30







Acknowledgments


References

























1 Introduction








[119865 (119905)]119896

= sum119899gt0

119865 (119899 119896) 119905119899

(1)





119860Δ

(119899 119896) = 120572119896

119865Δ

(119899 119896) (2)



119860Δ

(119899 119896) = 120572119899

119865Δ

(119899 119896) (3)


119891(119899)119905119899 119877(119905) = sum119899gt0

119903(119899)119905119899


119860Δ

(119899 119896) =

119899

sum119898=119896

119865Δ

(119899119898)119866Δ

(119898 119896) (4)


119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899



119886(119899)119905119899

are

119886 (119899) =

119899

sum119896=1

119865Δ

(119899 119896) 119903 (119896)

119886 (0) = 119903 (0)

(5)




119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899


119899ge0119886(119899)119905119899 are

119886 (119899) =

119899

sum119896=0

119891 (119896) 119903 (119899 minus 119896) (6)




119910119899(119909) =

119899

sum119896=0

(119899 + 119896)

(119899 minus 119896)119896(119909

2)119896

(7)


119901119899(119909) = 119909

119899

119910119899minus1

(1

119909) (8)


119901119899(119909) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)(1

2)119899minus119896

119909119896 (9)


sum119899=0

119901119899(119909)

119899119905119899


(10)




119862Δ

(119899 119896) =119896

119899(2119899 minus 119896 minus 1

119899 minus 1) (11)



2)

119861 (119905) = 119890119905

=

infin

sum119899=0

1

119899119905119899

(12)


119860Δ

(119899 119896) = (2119909)119896

(1

2)119899

119862Δ

(119899 119896) = 2119896minus119899

119909119896119896

119899(2119899 minus 119896 minus 1

119899 minus 1)

(13)


119899=0119887(119899)119905119899

are equal to

119887 (119899) =1

119899 (14)


119901119899(119909) = 119899

119899

sum119896=1

119860Δ

(119899 119896) 119887 (119896) =

119899

sum119896=1

(2119899 minus 119896 minus 1)


119909119896

(15)




119898119899+1

(119909 120573 119888) =(119888 minus 1)

119899

119888119899

sdot ((119909 minus 119887119899)119898119899(119909 120573 119888) minus 120582

119899119898119899minus1

(119909 120573 119888))

(16)

where

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) = 119909 minus 119887

0

119887119899=

(1 + 119888) 119899 + 120573119888

1 minus 119888

120582119899=

119888119899 (119899 + 120573 minus 1)

(1 minus 119888)2

(17)


1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) =

119909 (119888 minus 1) + 120573119888

119888

1198982(119909 120573 119888)

=1199092 (119888 minus 1)

2

+ 119909 ((2120573 + 1) 1198882 minus 2120573119888 minus 1) + (120573 + 1) 1205731198882

1198882

(18)


infin

sum119899=0

119898119899(119909 120573 119888)

119899119905119899

= (1 minus119905

119888)119909


(19)





119860 (119905) = (1 minus119905

119888)119909

=

infin

sum119899=0

(119909

119899) (minus1)

119899

119888minus119899

119905119899


=

infin

sum119899=0


119899) (minus1)

119899

119905119899

(20)


suminfin

119899=0119886(119899)119905119899 and 119861(119905) = sum

infin


follows

119886 (119899) = (119909

119899) (minus1)

119899

119888minus119899

119887 (119899) = (minus119909 minus 120573

119899) (minus1)

119899

(21)


119898119899(119909 120573 119888) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896)

= (minus1)119899

119899

119899

sum119896=0

(119909

119896)(


119899 minus 119896) 119888minus119896

(22)



119872119899+1

(119909 120575 120578) = (119909 minus 119887119899)119872119899(119909 120575 120578) minus 120582

119899119872119899minus1

(119909 120575 120578)

(23)

where1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 119887

0

119887119899= (2119899 + 120578) 120575

120582119899= (1205752

+ 1) 119899 (119899 + 120578 minus 1)

(24)


1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 120575120578

1198722(119909 120575 120578) = 119909

2

minus 1199092120575 (1 + 120578) + 120578 ((120578 + 1) 1205752

minus 1)

(25)


infin

sum119899=0

119872119899(119909 120575 120578)

119899119905119899

= ((1 + 120575119905)2

+ 1199052

)minus1205782


1 + 120575119905))

(26)




119860 (119905) = ((1 + 120575119905)2

+ 1199052

)minus1205782

= (1 + 2120575119905 + (1205752

+ 1) 1199052

)minus1205782

119861 (119905) = exp (119909tanminus1 ( 119905

1 + 120575119905))

(27)


1(1198602(119905)) and we expand 119860


theorem

1198601(119905) = (1 + 119905)

minus1205782

=

infin

sum119899=0

(minus120578

2119899

) 119905119899

1198602(119905) = 2120575119905 + (120575

2

+ 1) 1199052

(28)


infin

119899=01198861(119899)119905119899

are

1198861(119899) = (

minus120578

2119899

) (29)


1198862119896minus119899

119887119899minus119896

(119896

119899 minus 119896) (30)


119860Δ

2(119899 119896) = (2120575)

2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896) (31)


infin

119899=0119886(119899)119905119899

119886 (0) = 1198861(0)

119886 (119899) =

119899

sum119896=1

119860Δ

2(119899 119896) 119886

1(119896)

(32)


119886 (0) = 1

119886 (119899) =

119899

sum119896=1

(2120575)2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896)(

minus120578

2119896

) (33)


1(1198612(119905)) where

1198611(119905) = 119890

119905

=

infin

sum119899=0

1

119899119905119899

1198612(119905) = 119909tanminus1 ( 119905

1 + 120575119905)

(34)



functions 1199091198621(1198622(119905)) where

1198621(119905) = tanminus1 (119905)

1198622(119905) =

119905

1 + 120575119905

(35)


((minus1)(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896] (36)

where [119895



119862Δ

1(119899 119896) = ((minus1)

(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

sdot

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896]

(37)


(119899 minus 1

119896 minus 1) 119886119896

119887119899minus119896

(38)


to

119862Δ

2(119899 119896) = (

119899 minus 1

119896 minus 1) (minus120575)

119899minus119896

(39)


2(119905)

119861Δ

2(119899 119896) = 119909

119896

119899

sum119898=119896

119862Δ

2(119899119898) 119862

Δ

1(119898 119896) (40)


infin

119899=01198871(119899)119905119899

are defined by

1198871(119899) =

1

119899 (41)


infin

119899=0119887(119899)119905119899

119887 (0) = 1198871(0)

119887 (119899) =

119899

sum119896=1

119861Δ

2(119899 119896) 119887

1(119896)

(42)


119887 (0) = 1

119887 (119899) =

119899

sum119896=1

119909119896

119899

119899

sum119898=119896

(119899

119898)120575119899minus119898

(1 + (minus1)

119898+119896

(minus1)(119898+119896)2minus119899

)

sdot

119898

sum119895=119896


(119895 minus 1)(119898

119895)[

119895

119896]

(43)


119872119899(119909 120575 120578) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896) (44)



Acknowledgment


References





































119892 (119911) =1

119911+ 1198870+

infin

sum119899=1

119887119899119911119899 (1)




1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911) (2)






119899| le

(119861 minus 119860)(119899 + 1) for 119892 isin Σ[119860 119861]



ℎ (119908) = 119892minus1

(119908) =1

119908+

infin

sum119899=0

119861119899119908119899

=1


1

119899119870119899

119899+1(1198870 1198871 119887

119899) 119908119899

(3)

where 119908 isin D

119870119899

119899+1(1198870 1198871 119887

119899)

= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)


3119887119899minus4

0(1198874+ 311988711198872)

+ sum119895ge5

119887119899minus119895

0119881119895

(4)


variables 1198871 1198872 119887

119899





119899|



119899| and




119899=1119901119899119911119899 be so that 119877119890(119901(119911)) gt 0


100381610038161003816100381610038161199012+ 1205721199012

1

10038161003816100381610038161003816le 2 + 120572

1003816100381610038161003816119901110038161003816100381610038162

(5)



119899=1119888119899119911119899


100381610038161003816100381610038161198882+ 1205741198882

1

10038161003816100381610038161003816le 1 + (120574 minus 1)

1003816100381610038161003816119888110038161003816100381610038162

(6)

Proof Write

119901 (119911) =[1 + 120593 (119911)]

[1 minus 120593 (119911)] (7)

where 119901(119911) = 1+suminfin



1= 21198881and 119901

2=

2(1198882+ 11988821)


2= 2(1198882+ 11988821) in (5) we

obtain100381610038161003816100381610038162 (1198882+ 1198882

1) + 120572 (2119888

1)210038161003816100381610038161003816le 2 + 120572

10038161003816100381610038162119888110038161003816100381610038162 (8)

or100381610038161003816100381610038161198882+ (1 + 2120572) 119888

2

1

10038161003816100381610038161003816le 1 + 2120572

1003816100381610038161003816119888110038161003816100381610038162

(9)




(119894)10038161003816100381610038161198870

1003816100381610038161003816 le

119861 minus 119860


119861 minus 119860 119900119905ℎ119890119903119908119894119904119890

(119894119894)10038161003816100381610038161198871

1003816100381610038161003816

le

119861 minus 119860


2 (119861 minus 119860)|1198870|2

119894119891 0 le 119860 le 1 minus 2119861

119861 minus 119860

2 119900119905ℎ119890119903119908119894119904119890

(119894119894119894)100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2

(119894V) 10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 119894119891 119887

119896= 0 119891119900119903 0 le 119896 le 119899 minus 1

(10)


1199111198921015840 (119911)

119892 (119911)= minus1 minus

infin

sum119899=0

119865119899+1

(1198870 1198871 1198872 119887

119899) 119911119899+1

(11)

where 119865119899+1

(1198870 1198871 1198872 119887




5=



119865119899+1

(1198870 1198871 119887

119899)


119899+1=119899+1

119860 (1198941 1198942 119894

119899+1) 1198871198941

01198871198942

1sdot sdot sdot 119887119894119899+1

119899

(12)

where

119860 (1198941 1198942 119894

119899+1)

= (minus1)(119899+1)+2119894

1+sdotsdotsdot+(119899+2)119894

119899+1

(1198941+ 1198942+ sdot sdot sdot + 119894

119899+1minus 1) (119899 + 1)

(1198941) (1198942) sdot sdot sdot (119894

119899+1)

(13)



119908ℎ1015840 (119908)


infin

sum119899=1

119865119899+1

(1198610 1198611 1198612 119861

119899) 119908119899+1

(14)

where119865119899+1

(1198610 1198611 1198612 119861


119899 + 1 given by

119865119899+1

= minus119899 (119899 minus (119899 + 1))

119899 (119899 minus 2119899)119861119899

0

minus119899 (119899 minus (119899 + 1))


01198611

minus119899 (119899 minus (119899 + 1))


01198612

minus119899 (119899 minus (119899 + 1))


0

sdot (1198613+119899 minus (2119899 minus 3)

21198612


119861119899minus119895

0119870119895

(15)


variables 1198611 1198612 119861

119899minus1and

1198610= minus1198870 119861

119899= minus

1

119899119870119899

119899+1(1198870 1198871 119887

119899) (16)


1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198881 1198882 119888

119899 119861) 119911119899

(17)

119908ℎ1015840 (119908)

ℎ (119908)= minus

1 + 119860120595 (119908)

1 + 119861120595 (119908)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198891 1198892 119889

119899 119861) 119908

119899

(18)


119899=1119888119899119911119899 and 120595(119908) = sum

infin

119899=1119889119899119908119899 are two



119899(1198961 1198962 119896

119899 119861) are given by

119870119901

119899(1198961 1198962 119896

119899 119861)

=119901

(119901 minus 119899)119899119896119899

1119861119899minus1

+119901


11198962119861119899minus2

+119901


11198963119861119899minus3

+119901


1

sdot [1198964119861119899minus4

+119901 minus 119899 + 3

21198962

2119861]

+119901


1

sdot [1198965119861119899minus5

+ (119901 minus 119899 + 4) 11989621198963119861]

+ sum119895ge6

119896119899minus119895

1119883119895

(19)


variables 1198962 1198963 119896

119899


minus119865119899+1

(1198870 1198871 1198872 119887

119899) = (119860 minus 119861)119870

minus1

119899+1(1198881 1198882 119888

119899+1 119861)

(20)


minus 119865119899+1

(1198610 1198611 1198612 119861

119899)

= (119860 minus 119861)119870minus1

119899+1(1198891 1198892 119889

119899+1 119861)

(21)


1198870= (119860 minus 119861) 119888

1

minus1198870= (119860 minus 119861) 119889

1

(22)

21198871minus 1198872

0= (119861 minus 119860) (119888

2

1119861 minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (119889

2

1119861 minus 1198892)

(23)




0and then adding

them gives

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus 1198611198882

1minus 1198611198892

1) (24)



21198872

0= (119861 minus 119860) (119888

2+ 1198892minus

2119861

(119861 minus 119860)21198872

0) (25)


sides we obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(119861 minus 119860)

2

(1198882+ 1198892)

2 (2119861 minus 119860)le

119861 minus 119860

radic2119861 minus 119860 (26)



119861 minus 119860



we obtain

41198871= (119861 minus 119860) [(119889

2minus 1198882) minus (119889

2

1minus 1198882

1) 119861] = (119861 minus 119860) (119889

2minus 1198882)

(28)


100381610038161003816100381611988711003816100381610038161003816 le

119861 minus 119860

2 (29)


21198871= (119861 minus 119860) (119889

2+ 1198601198892

1minus 2119861119889

2

1) (30)


100381610038161003816100381611988711003816100381610038161003816 le

1

2(119861 minus 119860) (

100381610038161003816100381610038161198892+ 1198601198892

1

10038161003816100381610038161003816+ 2119861

1003816100381610038161003816119889110038161003816100381610038162

)

le1

2(119861 minus 119860) (1 + (119860 minus 1)

1003816100381610038161003816119889110038161003816100381610038162

+ 211986110038161003816100381610038161198891

10038161003816100381610038162

)

=119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

(31)




2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

lt119861 minus 119860

2 (32)


21198871minus 1198872

0= (119861 minus 119860) (

119861

(119861 minus 119860)21198872

0minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (

119861

(119860 minus 119861)21198872

0minus 1198892)

(33)


100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2 (34)


(119899 + 1) 119887119899= (119860 minus 119861) 119888

119899+1 (35)


obtain

10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 (36)







0|2 given by






References



























1 Introductions


119891 (119911) = 119911 +

infin

sum120581=2

119886120581119911120581

(1)


119867[119886 119899] = 119901 isin 119867 119901 (119911) = 119886 + 119886119899119911119899

+119886119899+1

119911119899+1


(2)


119865 (119911) = 119911120573

+

infin

sum120581=2

120582120581119886120581119911120573+120581minus1

120573 isin N

120582120581= 120573120581minus1

119911 =1

120573 120573 |119911| lt 1

(3)


(119891 lowast 119892) (119911) =

infin

sum119896=0

119886119896119887119896119911119896

(4)


119896=0119886119896119911119896 and 119892(119911) = sum

infin

119896=0119887119896119911119896 are analytic in

119880

Let 120595(119911) = 119911120573 + sum

infin



120595 (119911) lowast119911120573

1 minus 120573119911 119911 =

1

120573 120573 isin N 120573 |119911| lt 1 (5)



119876(120485) (120572 120591 120574 120573)

= 119865 (119911) isin 119867 (120573) R119890

times120572[119865(119911)]

(120485)

[119911120573](120485)

+120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

gt 120574 119911 isin 119880

(6)

where

119911 = 0 120573 gt 120485 120485 isin N cup 0 also 120572 + 120591 = 120574 (7)







119891 (119911) =1

120572 + 120591int|119909|


times

infin

sum120581=0

(120572 + 120591) 119909120581

119911120581+1

(120581 + 1) 120591 + 120572] 119889120583 (119909)

(8)




Θ120572120573

(119911) = (120573int119911

0

119905120573minus120572minus1

[119891 (119905)]120572

119892 (119905) 119889119905)

1120573

(10)






119901 (119911) = int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909) (|119911| lt 1) (11)



119892 (119911) +1199111198921015840 (119911)

119888≺ ℎ (119911) (12)

then

119892 (119911) ≺ 119902 (119911) =119888119911minus119888120581

120581int119911

0

119905(119888minus120581)120581

ℎ (119905) 119889119905 ≺ ℎ (119911) (13)



1199111199011015840 (119911)

119901 (119911)≺ ℎ (119911) (14)

then

119901 (119911) ≺ 119902 (119911) = 119886 exp [1

119899int119911

0

ℎ (120585) 120585minus1

119889120585] (15)




(ii) |1199011015840(119911)| le 2R119890119901(119911)(1 minus 1199032)


100381610038161003816100381610038161003816100381610038161003816

1199111199011015840 (119911)

119901 (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 119901 (119911)

(1 minus 119903)2

(16)




(120485)

(120572120591120574120573)




119911120485

[119865 (119911)](120485)

=

120573

119875120485

(120572 + 120591)int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120573 minus 120485) (120572 + 120591) 119909120581119911120573+120581

(120572 + 120591) (120573 minus 120485) + 120581120591] 119889120583 (119909)

(17)

where 119899119875119903




119901 (119911)

=

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

isin 119867 [1 119899]

(18)

ByTheorem 3

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

= int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909)

(19)


[119865 (119911)](120485+1)

[119911120573](120485+1)

+120572[119865 (119911)]

(120485)

120591[119911120573](120485)

=1

120591int|119909|=1

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909119911

1 minus 119909119911119889120583 (119909)

(20)

which yields

119911((120572+120591)(120485minus120573)+120591)120591

int119911

0

[[119865 (120585)]

(120485+1)

[120585120573](120485+1)

+120572[119865 (120585)]

(120485)

120591[120585120573](120485)

]


119889120585

=1

120591int|119909|=1

119911((120572+120591)(120485minus120573)+120591)120591

times int119911

0

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909120585

1 minus 119909120585


119889120585 119889120583 (119909)

(21)



shows that 119865(119911) isin 119876(120485)(120572 120591 120574 120573)



119865 (119911) =1

120572 + 120591int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581119911120573+120581

120572120573 + 120591 (120581 + 120573)] 119889120583 (119909)

(22)


1




10038161003816100381610038161198861205811003816100381610038161003816 le 119867 (120572 120591 120574 120573) (23)

where

119867(120572 120591 120574 120573) =2 (120572 + 120591 minus 120574) (120573 minus 120485) [

120573

119875120485]

120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(24)


119911120485

[119865119909(119911)](120485)

120573119875120485

= 119911120573

+

infin

sum120581=2


(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)(|119909| = 1)

(25)




119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [[120581 minus 1] 119903

120581minus1]

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

le2119903R119890 Ψ (119911)

(1 minus 119903)2

(26)


119866 (119911) equiv120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

+120572[119865(119911)]

(120485)

[119911120573](120485)

minus 120574 (27)

and then119911Ψ1015840 (119911)

Ψ (119911)=

suminfin

120581=2120599 (120572 120591 120574 120573) [120581 minus 1] 119886

120581119911120581minus1

1 + suminfin

120581=2120599 (120572 120591 120574 120573) 119886

120581119911120581minus1

(28)

where

120599 (120572 120591 120574 120573) =120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591 minus 120574) [120573

119875(120485)

]

(29)

From (23) and (28) we got100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [120581 minus 1] 119903

120581minus1

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

(120581 ge 2) (30)


119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 Ψ (119911)

(1 minus 119903)2

(31)

Sinceinfin

sum120581=2


lt 2119903R119890 Ψ (119911)

1 +

infin

sum120581=2

2 (120573 minus 120485) 119903120581minus1

gt (1 minus 119903)2

(32)




(120485)

(120572 120591 120574 120573)



1199111198661015840 (119911)

119866 (119911) ≺ ℎ (119911) (33)

then

119866 (119911) ≺ 119902 (119911) = (120572 + 120591 minus 120574) exp [119899minus1

int119911

0

ℎ (120585) 120585minus1

119889120585 ] (34)



119866 (119911) = (120572 + 120591 minus 120574) +

infin

sum120581=2

120582120581[ [120573+120581minus1]119875

120485+1] 119886120581119911120581minus1

120573119875120485+1

(35)


1199111198661015840

(119911)

119866 (119911) ≺ ℎ (119911) (36)




1199111198661015840 (119911)

119866 (119911) ≺

radic119911

1 minus 119911(37)

then

119866 (119911) ≺ (120572 + 120591 minus 120574) 1 + radic119911

1 minus radic119911

1119899

120572 + 120591 = 120574 (38)




120573 gt 120485 + 1 with 120572 + 120591 = 0 If

119866 (119911) ≺ (120572 + 120591)1 + 119860119911

1 + 119861119911 (39)

then[119865(119911)]

(120485)

[119911120573](120485)

≺(120572 + 120591) (120573 minus 120485)

120591120581119911(120572+120591)(120485minus120573)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(40)


1

(120572 + 120591)119866 (119911) = 1 +

infin

sum120581=2

119862120581119886120581119911120581minus1

(41)

where

119862120581=

120582120581[ (120573+120581minus1)119875

120485+1] sdot [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591) [120573

119875120485+1

] (42)

Let 119892(119911) = [119865(119911)](120485)

[119911120573](120485) then 119892(119911) = 1 + sum

infin

120581=2120603120581119886120581119911120581minus1

where 120603120581= 120582120581[(120573+120581minus1)

119875120485+1

][(120573)

119875120485] and from (39) we have

119892 (119911) +120591

(120572 + 120591) (120573 minus 120485)1199111198921015840

(119911) ≺ (1 + 119860119911

1 + 119861119911) = ℎ (119911) (43)


119892 (119911) ≺(120572 + 120591) (120573 minus 120485)

120591120581119911minus(120572+120591)(120573minus120485)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(44)


Corollary 16 Let

119876(0)

(120572 120591 0 120573) =120572 (119865 (119911))

119911120573+

120591[119865 (119911)]1015840

[119911120573]1015840

(45)

If

119876(0)

(120572 120591 0 120573) ≺ (120572 + 120591) (1 + 119860119911

1 + 119861119911) (46)

then

119865 (119911)

119911120573≺

120572120573 + 120591 (120573 minus 120581)

120591120581119911minus120573(120572+120591)120591120581

times int119911

0

120585(120573(120572+120591)minus120591120581)120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(47)






Acknowledgments



References





















1 Introduction


119891 (119911) =119892 (119911)

ℎ (119911) (1)






R ℎ (119911) gt 0 (119911 isin 119880lowast

) (2)


119891 ≺ 119892 in 119880 or 119891 (119911) ≺ 119892 (119911) (119911 isin 119880lowast

) (3)


|119908 (119911)| le |119911| 119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880lowast

) (4)


119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0)

119891 (119880) sube 119892 (119880) (119911 isin 119880lowast

) (5)





2 Preliminaries


119891 (119911) =1

119911+

infin

sum119899=1

119886119899119911119899

(6)




119891119895(119911) =

1

119911+

infin

sum119899=1

119886119899119895119911119899

(7)


and 1198912(119911) by

(1198911lowast 1198912) =

1

119911+

infin

sum119899=1

11988611989911198861198992119911119899

(8)


119902120582120583

(119911) =1

119911+

infin

sum119899=1

(120582

119899 + 1 + 120582)

120583

119911119899

(120582 gt 0 120583 ge 0) (9)



119871120582120583119891 (119911) = (119891 (119911) lowast 119902

120582120583(119911))

=1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(10)


119905119886119892 and 119871


tively by

119871120582120583119892 (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198871198991003816100381610038161003816 119911119899


119871120582120583ℎ (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198881198991003816100381610038161003816 119911119899


ℎ (119911) isin Σ)

(11)


119871120582120583

(119891 lowast 119892) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast)

119871120582120583

(119891 lowast ℎ) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198881198991003816100381610038161003816 119911119899


)

(12)






property

120588119871120582120583

(119891 lowast 119892) (119911)

119871120582120583

(119891 lowast ℎ) (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911 (119911 isin 119880

lowast

) (13)


and 119871(120582 120583)(119891 lowast ℎ)(119911) = 0


120601 (120572 120573 119911) =1

119911+

infin

sum119899=0

(120572)119899+1

(120573)119899+1

119911119899

(14)



120601 (120572 120573 119911) =1

119911 21198651(1 120572 120573 119911) (15)

where

21198651(119887 120572 120573 119911) =

infin

sum119899=0

(119887)119899(120572)119899

(120573)119899

119911119899

119899(16)


120582120583(119911)


119871 (120572 120573 120582 120583) 119891 (119911) = (119891 (119911) lowast 120601 (120572 120573 119911) lowast 119902120582120583

(119911))

=1

119911+

infin

sum119899=1

(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(17)




119911(119871 (120572 120573 120582 120583) 119891 (119911))1015840

= 120572119871 (120572 + 1 120573 120582 120583) 119891 (119911)

minus (120572 + 1) 119871 (120572 120573 120582 120583) 119891 (119911) (18)


120588119871 (120572 + 1 120573 120582 120583) 119891 (119911)

119871 (120572 120573 120582 120583) 119891 (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911

(0 le 119861 lt 119860 le 1 120588 gt 0)

(19)



relation (19) above






infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861)

minus10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(20)


119891119899(119911) =

1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

(21)



1003816100381610038161003816100381610038161003816100381610038161003816

120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899) 119911119899+1

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899119911119899+1

)

minus11003816100381610038161003816100381610038161003816100381610038161003816

le 120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899)10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899

10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816)

minus1

le 1

(22)





100381610038161003816100381610038161003816100381610038161003816lt 1

(119911 isin 119880lowast)

(23)


Σ120583




120582(120588 119860 119861) then

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119899 ge 1

(24)


(21)


120582(120588 119860 119861)



1

119903minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903

le1003816100381610038161003816119891 (119911)

1003816100381610038161003816

le1

119903+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903


1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

le100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(25)



inequalityinfin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(26)


1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

le1

119903+ 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

119903+ 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|minus

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

ge1

119903minus 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

ge1

119903minus 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(27)


sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(119860 minus 119861) (2 + 120582)

120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(28)

Hence

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2+

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

le1

1199032+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

ge1

1199032minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816

ge1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(29)



120582(120588 119860 119861) which




disk |119911| lt 1199031 where

1199031= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(30)



119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (31)

For |119911| lt 1199031 we have

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 1 + 120582))

120583

119886119899119911119899

1119911 + suminfin

119899=1(120582(119899 + 1 + 120582))

120583

119886119899119911119899

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

100381610038161003816100381610038161003816100381610038161003816

lesuminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(32)


sum119899=1

(119899 + 1) (120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1


infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

)

(33)

or

suminfin

119899=1(119899 + 2 minus 120575) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(1 minus 120575)le 1 (34)


(119899 + 2 minus 120575) (120582(119899 + 1 + 120582))120583

|119911|119899+1

(1 minus 120575)

le (120582

119899 + 120582 + 1)

120583

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899 ge 1

(35)



|119911| le (1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899+1

(36)




disk |119911| lt 1199032 where

1199032= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(37)



1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891 (119911))10158401015840

(119891 (119911))1015840+ 2

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (38)






infin

sum119899=1

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(39)


119891119899(119911)

=1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

119899 ge 1

(40)




120582(120588 120572 120573 119860 119861) then1003816100381610038161003816119886119899

1003816100381610038161003816

le(119860 minus 119861) (119899 + 120582 + 1)

120583

(120573)119899+1

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861)) (120572)

119899+1

119899 ge 1

(41)


(40)




1199033= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(42)





1199034= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(43)






Acknowledgment


References






















1 Introduction


Φ (119911 119904 119886) =

infin

sum119899=0

119911119899

(119899 + 119886)119904


0 119904 isin C when |119911| lt 1

R (119904) gt 1 when |119911| = 1)

(1)


119904(120585) defined by

120577 (119904) =

infin

sum119899=1

1

119899119904= Φ (1 119904 1) = 120577 (119904 1) (R (119904) gt 1)

120577 (119904 119886) =

infin

sum119899=0

1

(119899 + 119886)119904


0)

ℓ119904(120585) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 1)119904= Φ (e2120587119894120585 119904 1) (R (119904) gt 1 120585 isin R)

(2)


119904(119911)

Li119904(119911) =

infin

sum119899=1

119911119899

119899119904= 119911Φ (119911 119904 1)


(3)


120601 (120585 119886 119904) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 119886)119904= Φ (e2120587119894120585 119904 119886)



when 120585 isin Z)

(4)




Φ (119911 119904 119886) =1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

1 minus 119911eminus119905d119905

(R (119886) gt 0R (119904) gt 0 when |119911| ≦ 1 (119911 = 1) R (119904) gt 1

when 119911 = 1) (5)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) =

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

119911119899

(119899 + 119886)119904

(119901 119902 isin N0 120582

119895isin C (119895 = 1 119901)

119886 120583119895isin C Z

minus

0(119895 = 1 119902)

120588119895 120590

119896isin R

+

(119895 = 1 119901 119896 = 1 119902)





lowast

)

(6)

with

nablalowast

= (

119901

prod119895=1

120588minus120588119895

119895) sdot (

119902

prod119895=1

120590120590119895

119895)

Δ =

119902

sum119895=1

120590119895minus

119901

sum119895=1

120588119895

Ξ = 119904 +

119902

sum119895=1

120583119895minus

119901

sum119895=1

120582119895+119901 minus 119902

2

(7)



(120582)120581=Γ (120582 + 120581)

Γ (120582)

=


1 (120581 = 0 120582 isin C 0)

(8)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0)

(9)



119898or

119897Ψ119898(119897 119898 isin N

0) involved


119897119865119898(119897 119898 isin N




1198871 119887

119898such that

119886119895isin C (119895 = 1 119897) 119887

119895isin C Z

minus

0(119895 = 1 119898)

(10)


119897Ψlowast

119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

=

infin

sum119899=0

(1198861)1198601119899sdot sdot sdot (119886

119897)119860119897119899

(1198871)1198611119899sdot sdot sdot (119887

119898)119861119898119899

119911119899

119899

=Γ (119887


119898)

Γ (1198861) sdot sdot sdot Γ (119886

119897) 119897Ψ119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

(119860119895gt 0 (119895 = 1 119897) 119861

119895gt 0 (119895 = 1 119898)

1 +

119898

sum119895=1

119861119895minus

119897

sum119895=1

119860119895≧ 0)

(11)


|119911| lt nabla = (

119897

prod119895=1

119860minus119860119895

119895) sdot (

119898

prod119895=1

119861119861119895

119895) (12)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

Γ (119904)intinfin

0


119905120582)

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0)

(13)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 0 120582)

= Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

= e119887Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 0)

(14)


119901 minus 1 = 119902 = 0 (1205821= 120583 120588

1= 1) (15)


Θ120582

120583(119911 119904 119886 119887)

=1

Γ (119904)intinfin

0


119905120582) (1 minus 119911eminus119905)

minus120583

d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0 120583 isin C)

(16)


R (119904) gt 0 when 119887 = 0 |119911| ≦ 1 (119911 = 1) (17)

or

R (119904 minus 120583) gt 0 when 119887 = 0 119911 = 1 (18)




Θ120582


Θ120582

120583(119911 119904 119886 0) = Φ

lowast

120583(119911 119904 119886) =

1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

(1 minus 119911eminus119905)120583d119905

(R (119886) gt 0 R (119904) gt 0 when |119911| ≦ 1 (119911 = 1)

R (119904 minus 120583) gt 0 when 119911 = 1)

(19)



[4] as follows

Φlowast

120583(119911 119904 119886) =

infin

sum119899=0

(120583)119899

(119886 + 119899)119904

119911119899

119899 (20)


120583(119911 119904 119886 119887) that is


Θ1

1(1 119904 119886 119887)

= 120577119887(119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1


119905) d119905

(21)




(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899

119899(120582 gt 0)

(22)



119867119898119899

pq (119911)

= 119867119898119899

pq [119911 |(119886

1 119860

1) (119886p 119860p)

(1198871 119861

1) (119887q 119861q)

]

=1

2120587119894intL

Ξ (119904) 119911minus119904d119904 (119911 isin C 0

1003816100381610038161003816arg (119911)1003816100381610038161003816 lt 120587)

(23)

where

Ξ (119904) =prod

119898

119895=1Γ (119887

119895+ 119861

119895119904) sdot prod

119899

119895=1Γ (1 minus 119886

119895minus 119860

119895119904)

prodp

119895=119899+1Γ (119886

119895+ 119860

119895119904) sdot prod

q

119895=119898+1Γ (1 minus 119887

119895minus 119861

119895119904)

(24)


119895gt 0 (119895 =

1 p) 119861119895gt 0 (119895 = 1 q) 119886

119895isin C (119895 = 1 p)



Γ (119887119895+ 119861

119895119904)

119898

119895=1

(25)


Γ (1 minus 119886119895+ 119860

119895119904)

119899

119895=1

(26)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902





1 minus 119905)

=

infin

sum119899=0

119871(120572)

119899(119909) 119905

119899

(|119905| lt 1 120572 isin C)

(27)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=eminus119887

Γ (119904)

infin

sum119899=0

119899

sum119896=0

(minus1)119896

(119899

119896) sdot Γ (119904 + 120582 (120572 + 119896 + 1))

sdot 119871(120572)

119899(119887)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 + 120582 (120572 + 119895 + 1) 119886)

(R (119886) gt 0 R (119904 + 120582120572) gt minus120582)

(28)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) (29)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



D120572

119911119911

119901

119891 (119911) =1

Γ (minus120572)int119911

0

119891 (120585) 120585119901


(R (120572) lt 0 R (119901) gt 1)

(30)


D120572

119911119911

119901

119891 (119911) =d119898

d119911119898D

120572minus119898

119911119911

119901

119891 (119911) (31)



Im(120585)

Branch line for

Branch line for


exp[p(ln(g(120585)))]

C1

C2

C3

C4

Re(120585)

z

a

gminus1(0)





119901


119863120572

119892(119911)[119892 (119911)]

119901

119891 (119911)

=eminus119894120587119901Γ (1 + 120572)4120587 sin (120587119901)

times int119862(119911+119892

minus1(0)+119911minus119892

minus1(0)minus119865(119886)119865(119886))

sdot119891 (120585) [119892 (120585)]

119901

1198921015840 (120585)

[119892 (120585) minus 119892 (119911)]120572+1

d120585

(32)



1cup119862

2cup119862

3cup119862








119892(119911)[119892(119911)]

119901



119863120572

119911119911

119901

=Γ (1 + 119901)

Γ (1 + 119901 minus 120572)119911119901minus120572

(R (119901) gt minus1) (33)



11205881199011205901120590119902)

12058211205821199011205831120583119902


119863120572

119911119911

120573minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899)

1198871120582

| (119904 1) (01

120582)]119863

120572

119911119911

120573minus1119911119899

119899

=Γ (120573)

120582Γ (119904) Γ (120573 minus 120572)

infin

sum119899=0

(120573)119899

(120573 minus 120572)119899

sdot

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899+120573minus120572minus1

119899

=Γ (120573)


Φ(1205881120588119901112059011205901199021)

12058211205821199011205731205831120583119902120573minus120572

(119911 119904 119886 119887 120582)







119863120572

119892(119911)119891 (119911) = 119863

120572

ℎ(119911)119891 (119911) 119892

1015840(119911)

ℎ1015840(119911)(ℎ(119908) minus ℎ(119911)

119892(119908) minus 119892(119911))

120572+1

100381610038161003816100381610038161003816100381610038161003816119908=119911

(35)




119911119901119874120572

120573119891 (119911)

= 119901(119911119901minus1

)minus120572

sdot119911

119874120572

120573119891 (119911) (119911

119901minus1

)120572

119901minus1

prod119904=1

(1 minus119911

119908eminus(2120587119894119904)119901)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(36)

where

119892(119911)119874120572


Γ (120573)

Γ (120572)[119892 (119911)]

1minus120573

119863120572minus120573

119892(119911)[119892 (119911)]

120572minus1

sdot sdot sdot

(37)



120579 (119911) = (119911 minus 1199110) 119902 (119911) (38)


119870

=0 1 [119888] ([119888] 119905ℎ119890 119897119886119903119892119890119904119905 119894119899119905119890119892119890119903 119899119900119905 119892119903119890119886119905119890119903 119905ℎ119886119899 119888)

(39)


0and let

120596 = exp (2120587119894119886) (40)



sum119896isin119870

119888minus1

120596minus120574119896

119891 (120579minus1

(120579 (119911) 120596119896

))

=

infin


[120579 (119911)]119888119899+120574

Γ (119888119899 + 120574 + 1)

sdot119863119888119899+120574

119911minus119887119891 (119911) 120579

1015840

(119911) (119911minus119911

0

120579 (119911))

119888119899+120574+1

100381610038161003816100381610038161003816100381610038161003816119911=1199110

(1003816100381610038161003816119911minus1199110

1003816100381610038161003816=100381610038161003816100381611991101003816100381610038161003816)

(41)



119891 (119911) = 119888

infin


(119911 minus 1199110)119888119899+120574

Γ(119888119899 + 120574 + 1)119863

119888119899+120574

119911minus119887119891(119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199110

(42)



1)(119911minus119911

2)) where 119911

1and 119911

2are two



120596 = exp(2120587119894119886) (43)


1and 119911


curves

119862 (119905) 119862 (119905) subR 0 lt 119905 ≦ 119903 (44)

be defined by

119862 (119905) = 1198621(119905) cup 119862

2(119905)

= 119911 1003816100381610038161003816120582119905 (1199111 1199112 119911)

1003816100381610038161003816 =1003816100381610038161003816100381610038161003816120582119905(119911

1 119911

21199111+ 119911

2

2)1003816100381610038161003816100381610038161003816

(45)

where

120582119905(119911

1 119911

2 119911) = [119911 minus

1199111+ 119911

2

2+ 119905 (

1199111minus 119911

2

2)]

sdot [119911 minus (1199111+ 119911

2

2) minus 119905 (

1199111minus 119911

2

2)]

(46)


1+ 119911



1199111+ 119911

2

2+ (

1199111minus 119911

2

2) 119905 (47)


2

2minus (

1199111minus 119911

2

2) 119905 (48)

Im(120585)

Branch line for

Branch line for

C1

C2

z1

z2

Re(120585)

z

c =z1 + z2

2

(120585 minus z2)120583120579(120585)

aminus120574minus1

(120585 minus z1)120579(120585)

aminus120574minus1



[(119911 minus 1199111) (119911 minus 119911

2)]

120582


2))))

(49)


plusmn

defined by

119871plusmn=

119911 119911 =1199111+ 119911

2

2plusmn 119905 (

1199111minus 119911

2

2) (0 ≦ 119905 ≦ 1)

119911 119911 =1199111+ 119911

2

2plusmn 119894119905 (

1199111minus 119911

2

2) (119905 lt 0)

(50)



1)(119911minus119911

2) gt 0 (v)


2)119901


+cup 119871


119911minus1199112

(119911 minus 1199112)119901


119870 = 119896 119896 isin N arg(120582119905(119911

1 119911

21199111+ 119911

2

2))

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) +

2120587119896

119886

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) + 2120587

(51)


defined by

120585 =1199111+ 119911

2

2+1199111minus 119911

2


sum119896isin119870

119888minus1120596minus120574119896

1199111minus 119911

2

119891 (120601minus1

(120596119896

120601 (119911)))

times [120601minus1

(120596119896

120601(119911)) minus 1199111]][120601

minus1

(120596119896

120601(119911)) minus 1199112]120583

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

[120601 (119911)]119888119899+120574

(52)


where

120601 (119911) =119911 minus 119911

1

119911 minus 1199112

(53)



2)120583

1199111minus 119911

2

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

(119911 minus 119911

1

119911 minus 1199112

)

119888119899+120574

(54)


2) where 119911

1and 119911




sum119896isin119870

119888minus1

120596minus120574119896

[(1199112minus 119911

1+ radicΔ

119896

2)

120572

(1199111minus 119911

2+ radicΔ

119896

2)

120573

sdot 119891 (1199111+ 119911

2+ radicΔ

119896

2)


sdot (1199112minus 119911

1minus radicΔ

119896

2)

120572


2minus radicΔ

119896

2)

120573

119891(1199111+ 119911

2minus radicΔ

119896

2)]

=

infin



eminus119894120587119888(119899+1)[120579 (119911)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911minus1199112


(120579 (119911)

(119911minus1199112) (119911minus119911

1))


1205791015840

(119911)119891 (119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199111

(55)

where

Δ119896= (119911

1minus 119911

2)2

+ 4119881 (120596119896

120579 (119911))

119881 (119911) =

infin

sum119903=1

119863119903minus1

119911[119902 (119911)]

minus119903

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

119911119903

119903

120579 (119911) = (119911 minus 1199111) (119911 minus 119911

2) 119902 ((119911 minus 119911

1) (119911 minus 119911

2))

(56)


(119911 minus 1199111)(119911 minus 119911

2)) and 119911


119891 (119911) = 119888119911minus120573

(119911 minus 1199111)minus120572

times

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot119863minus120572+119888119899+120574

119911119911


(119911 + 119908 minus 1199111)119891(119911)

10038161003816100381610038161003816119911=1199111

(119908=119911)

(57)



119863120572

119911119911

119901+119902

119906 (119911) V (119911)

=

infin


(120572

120574 + 119899)119863

120572minus120574minus119899

119911119911

119901

119906 (119911)119863120574+119899

119911119911

119902V (119911) (58)




119863120572

119911119911

120572+120573minus1

119906 (119911) V (119911)


sdot

infin


119863120572+]+1minus119899119911



(60)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 (1120582))]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot 119865(119896minus1)



where 120582 gt 0 and 119865(119899)119863


119865(119899)

119863[119886 119887

1 119887

119899 119888 119909

1 119909

119899]

=

infin

sum1198981119898119899=0

(119886)1198981+sdotsdotsdot+119898

119899

(1198871)1198981

sdot sdot sdot (119887119899)119898119899

(119888)1198981+sdotsdotsdot+119898

119899

1199091198981

1


119909119898119899

119899

119898119899

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091198991003816100381610038161003816 lt 1)

(62)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119896

(119911119896minus1)120572

sdot119911119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) (119911119896minus1

)120572

sdot

119896minus1

prod119904=1

(1 minus119911

119908eminus2120587119894119904119896)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(63)




119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

(64)



119896

(119911119896minus1)120572

119911

119874120572

120573

times

1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119911119899+(119896minus1)120572

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


sdot((119911119908) eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119908=119911

=119896

120582Γ (119904) (119911119896minus1)120572

times

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981



sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

times119911

119874120572

120573119911


119896minus1+119899+(119896minus1)120572

=119896

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


times(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

sdotΓ (120573) Γ (119898


119896minus1+ 119899 + 119896120572)

Γ (120572) Γ (120573 + 1198981+ sdot sdot sdot + 119898

119896minus1+ 119899 + 119896120572)


=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 1120582)]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(119899+119896120572)1198981+sdotsdotsdot+119898

119896minus1

(1+120572minus120573)1198981



119896minus1

sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

(65)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


119911minus1198881198990(119911 minus 119911

0)119888119899

Γ (119888119899 + 1) Γ (1 minus 119888119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(1003816100381610038161003816119911 minus 1199110

1003816100381610038161003816 =100381610038161003816100381611991101003816100381610038161003816 120582 gt 0)

(66)


Proof Setting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


0 we

have

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


(119911 minus 1199110)119888119899

Γ (1 + 119888119899)

sdot119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

(67)


0| = |119911

0|


119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

=119911minus1198881198990

Γ (1 minus 119888119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(68)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(69)


119911 =1199111

2+1199111



Proof By taking 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 9 with 119911

2= 0 120583 = 120572 ] = 120573 and 0 lt 119888 ≦ 1 we find

that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888(119911 minus 1199111)minus120573

119911minus120572

1199111

sdot

infin



sdot (119911 minus 119911

1

119911)119888119899+120574

sdot119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

(71)



119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120572+120573minus1

1

Γ (120572 + 119888119899 + 120574)

Γ (120572 + 120573)

times Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582)

(72)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573+120574

(119911 minus 1199111)minus120572+120574

119911120573+120572minus2120574minus1

1

sdot

infin


sin [(120573 minus 119888119899 minus 120574) 120587] e119894120587119888(119899+1)

sin [(120573 + 119888 minus 120574) 120587] Γ (1 minus 120572 + 119888119899 + 120574)

sdot (119911(119911 minus 119911

1)

11991121

)

119888119899

Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdot [(119911 minus 1199111)Φ

(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (120573 minus 119888119899 minus 120574

120572 + 120573 minus 2119888119899 minus 2120574) 119911

1

sdotΦ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)]

(73)


119911 =1199111

2+1199111



Proof Putting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 10 with 119911

2= 0 0 lt 119888 ≦ 1 119902(119911) = 1 and 120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2) we find that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573

(119911 minus 1199111)minus120572

sdot

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

(75)


119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

= 119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

+ (119911 minus 1199111)119863

minus120572+119888119899+120574

119911


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120573+120572minus2119888119899minus2120574

1


Γ (1 + 120573 + 120572 minus 2119888119899 minus 2120574)

sdot Φ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (119911 minus 119911

1

1199111

)Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdotΦ(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582) )

(76)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

=Γ (120591) Γ (1 + ] minus 120591) sin (120574120587)

120587

sdot

infin


(minus1)119899

Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)


(77)



12058211205821199011205831120583119902


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=

infin


(] minus 120591120574 + 119899


119911]minus1

sdot 119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(78)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)



119911]minus1 =

Γ (])Γ (120591 + 120574 + 119899)

119911120591+120574+119899minus1

119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911minus120574minus119899

Γ (1 minus 120574 minus 119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(79)



Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

= (Γ (120591) Γ (1 + ] minus 120591) sin (120573120587)




sdotsin (120579120587)

sin [(120579 + 120574) 120587]

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(80)



120572 = ] minus 120591 119906 (119911) = 119911120591minus120573

V (119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(81)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


sdot

infin


119863]minus120591+120574+1minus119899119911




119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(82)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

119863]minus120591+120574+1minus119899119911


=Γ (] minus 120579 minus 119899)


119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (120579 + 119899)

Γ (1 + 120579 + 120574)

times 119911120579+120574

Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(83)





lim119887rarr0

11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)] = 120582Γ (119904)

(120582 gt 0)

(84)



Φ(120588112058811990113120590

112059011990213)

12058211205821199011205721205831120583119902120573

(119911 119904 119886)

=3Γ (120573) Γ (3120572)

Γ (120572) Γ (120573 + 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

(3120572)119899

(120573 + 120572)119899

119911119899

(119899 + 119886)119904


(85)


1198651[119886 119887

1 119887

2 119888 119909

1 119909

2]

=

infin

sum11989811198982=0

(119886)1198981+1198982

(1198871)1198981

(1198872)1198982

(119888)1198981+1198982

1199091198981

1

1198981

1199091198982

2

1198982

(max 100381610038161003816100381611990911003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 lt 1)

(86)




1= 120583 in



Θ120582

120583(119911 119904 119886 119887)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(111)

120583120572+119888119899+120574120572+120573(119911

1 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(87)


119911 =1199111

2+1199111





Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886)

=Γ (120591) Γ (1 + ] minus 120591)



times (Γ (1 + 120574 + 120579) sin [(] minus 120591 + 120573) 120587]


sdot

infin


Γ (]minus120579minus119899) Γ (120579+119899)Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120582

(119911 119904 119886)





References
























































Toka Diagana










1 Introduction




119886(119899 119896) and 119878



(119905)119899=

119899

sum119896=0

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

(119905 minus 119886)119896

(1)

(119905 minus 119886)119899

=

119899

sum119896=0

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(119905)119896 (2)


119904119886(119899 119896) =

1

119896[119889119896

119889119905119896(119905)V]

119905=119886

119878119886(119899 119896) =

1

119896[Δ119896

(119905 minus 119886)119899

]119905=0

(3)


119904119886(119899 + 1 119896) = 119904

119886(119899 119896 minus 1) + (119886 minus 119899) 119904

119886(119899 119896) (4)

119878119886(119899 + 1 119896) = 119878

119886(119899 119896 minus 1) + (119896 minus 119886) 119878

119886(119899 119896) (5)


119904119886(0 0) = 1 119904

119886(119899 0) = (119886)

119899 119904

119886(0 119896) = 0 119899 119896 = 0

119878119886(0 0) = 1 119878

119886(119899 0) = (minus119886)

119899

119878119886(0 119896) = 0 119899 119896 = 0

(6)


1+ 119883


119896of





119875 [119882 = 119899] =119896

(1 minus 120579)minus119904

(minus log (1 minus 120579))119896

120579119899

119899(minus1)

119899minus119896

119904minus119904(119899 119896)

119875 [119885 = 119899] =119896

119890119898 (119890119897 minus 1)119896

119897119899

119899(minus1)

119899minus119896

119878minus119898119897

(119899 119896)

(7)






119870(1)

119899119903= 119903 + [log(119899 minus 1


1

119903+ 119900 (1)] (8)


2 Explicit Formula



119865119899(119909) = prod

119899


1198651015840

119899(119909) = sum


119899minus1le119899

119894isinN11989911989511198952119895119899minus1

1198911015840

119894(119909)

119899minus1

prod119896=1

119891119895119896(119909) (9)


1 2

119889

119889120585(120585119895 + 119886)

1= 119895

119889

119889120585(120585119895 + 119886)

2= (120585119895 + 119886) 119895 + (120585119895 + 119886 minus 1) 119895

(10)


1

119895119899119889

119889120585(120585119895 + 119886)



119899minus1

prod119896=1

(120585 +119886 minus 119895

119896

119895) (11)



1

119895119899119889119896

119889120585119896(120585119895 + 119886)


119899minus119896le119899minus1

119896

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(12)


1

119895119899119889119898

119889120585119898(120585119895 + 119886)


119899minus119898le119899minus1

119898

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(13)

Then

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899


119899minus119898le119899minus1

119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(14)




119889

119889120585

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895) = sum



prod119902=1

(120585 +119886 minus 119894

119902

119895)

(15)

119894119902

isin 1198951 1198952 119895

119899minus119898 has ( 119899minus119898





119902=1(120585 + (119886 minus





119895119899minus119898minus1





119902=1(120585 + (119886 minus 119895


of (1119895119899)(119889119898+1119889120585119898+1)(120585119895 + 119886)119899 Thus we have

1

119895119899119889119898+1

119889120585119898+1(120585119895 + 119886)

119899



119898 + 1

119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)



119899minus119898

prod119902=1

(120585 +119886 minus 119895

119902

119895)

(16)

Lemma 2 Let 119904(119899 119896 119886) =

(1119896)lim120585rarr0

((sum119896

119895=0(minus1)

119896minus119895

( 119896119895) (119889119896119889120585119896)(120585119895 + 119886)

119899)119896)

Then


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (17)

Proof Using Lemma 1

119904 (119899 119896 119886)

=1

119896lim120585rarr0

((

119896

sum119895=0


(119896

119895) 119895119899


119899minus119896le119899minus1

119896

times

119899minus119896

prod119902=1

(120585 +119886 minus 119895

119902

119895)) (119896)

minus1

)

(18)



(120585 + (119886 minus 119895119902)119895) = (1119895119899minus119896)prod

119899minus119896

119902=1(120585119895 + 119886 minus 119895

119902)


1

119896lim120585rarr0

[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(120585119895 + 119886 minus 119895119902)]

]

=1

119896[

[

119896

sum119895=0


(119896

119895) 119895119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)]

]

(19)

which boils down to


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (20)

since

1

119896

119896

sum119895=0


(119896

119895) 119895119896

= 119878 (119896 119896) = 1 (21)



119904119886(119899 119896) = 119904 (119899 119896 119886) = sum


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902) (22)

Proof We know that


119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (23)



119899+1minus119896)







119899minus119896+1le119899

119899minus119896+1

prod119902=1

(119886 minus 119895119902) (24)

is equal to


119899minus119896+1le119899minus1

119899minus119896+1

prod119902=1

(119886 minus 119895119902) + (119886 minus 119899)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119886 minus 119895119902)

(25)

This implies that

119904 (119899 + 1 119896 119886) = 119904 (119899 119896 minus 1 119886) + (119886 minus 119899) 119904 (119899 119896 119886) (26)



119886(119899 119896)


120582 = (1205821ge 120582


119896) (27)


119894


14= 11989115

= 11989123

= 11989131

= 11989142

= 1 119891119894119895= 0 elsewhere

(1 le 119895 le 120582119894)

0 0 0 1 10 0 11 0 00 10

(28)



(119886119894)119894ge0



1198860 1198861 119886

ℎ then |119879119889119860(ℎ 119903)| = ( ℎ+1




120596-tableau is











119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)


120596 (|119888|) (29)




is equal to 119904119886(119899 119896)



0 1 2 119899 minus 1 and

120596119860(120601) = prod

119888isin120601

120596 (|119888|)

=

119899minus119896

prod119894=1

(119886 minus 119895119894) 119895

119894isin 0 1 2 119899 minus 1

(30)

If 119886 = 1198861+ 1198862for some 119886

1and 119886

2 then with 120596lowast(119895) = 119886

2minus 119895

120596119860(120601) =

119899minus119896

prod119894=1

(1198861+ 120596lowast

(119895119894))

=

119899minus119896

sum119903=0

119886119899minus119896minus119903

1sum


119903le119895119899minus119896

119903

prod119894=1

120596lowast

(119902119894)

(31)






may write

120596119860(120601) = sum

120595isin119861120601

120596119860(120595) (32)


120596119860(120595) = 119886

119899minus119896minus119903

1

119903

prod119894=1

120596lowast

(119902119894) (33)

where 1199021 1199022 119902

119903isin 119895

1 1198952 119895


written as

119904119886(119899 119896) = sum

120601isin119879119889119860(119899minus1119899minus119896)

sum120595isin119861120601

120596119860(120595) (34)



1 1198952 119895

119899minus119896



(119899

119896)(

119899 minus 119896

119903) (35)





( 119899119896) ( 119899minus119896

119903)

( 119899119903)

= (119899 minus 119903

119896) (36)


119904119886(119899 119896) =

119899minus119896

sum119903=0

(119899 minus 119903

119896) 119886119899minus119896minus119903

1sum120595isin119861119903


120596lowast

(|119888|) (37)



119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

1sum



120596lowast

(|119888|) (38)

Clearly 119861119899minus119895



tity

119904119886(119899 119896) =

119899

sum119895=119896

(119895

119896) 119886119895minus119896

11199041198862

(119899 119895) (39)


1and 119886

2





119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (40)






2isin 1198791198891198602(119895 minus 1 119895 minus 119899+ 119896)


2 we obtain an


119860

(119898 +


sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601)

=

119899

sum119896=0

sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011)

times

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)

(41)


Note that

sum

1206012isin1198791198891198602 (119895minus1119895minus119899+119896)

1205961198602

(1206012)


119895minus119899+119896le119898+119895minus1

119895minus119899+119896

prod119902=1

(119886 minus 119892119902)



119895minus119899+119896

prod119902=1

(119886 minus (119898 + 119892119902))

= 119904119886minus119898

(119895 119899 minus 119896)

(42)


sum

1206011isin1198791198891198601 (119898minus1119898minus119896)

1205961198601

(1206011) = 119904

119886(119898 119896)

sum

120601isin119879119889119860(119898+119895minus1119898+119895minus119899)

120596119860(120601) = 119904

119886(119898 + 119895 119899)

(43)

Thus

119904119886(119898 + 119895 119899) =

119899

sum119896=0

119904119886(119898 119896) 119904

119886minus119898(119895 119899 minus 119896) (44)



convolution formula

119904119886(119899 + 1119898 + 119895 + 1) =

119899

sum119896=0

119904119886(119896 119899) 119904

119886minus(119896+1)(119899 minus 119896 119895) (45)

Proof Let


in 1198601= 0 1 119896 minus 1


are in 1198602= 119896 + 1 119899








21199092 + sdot sdot sdot + 119886

119899119909119899 has only

real zeros then

1198862

119896ge 119886119896+1

119886119896minus1

119896

119896 minus 1

119899 minus 119896 + 1

119899 minus 119896(119896 = 2 119899 minus 1) (46)


sum119896=0

119904119886(119899 119896) (119905 + 119886)

119896

(47)



119896=0119904119886(119899 119896)119905119896 are 119886 119886minus1 119886minus



119899


and hence unimodal


⟨119905⟩119899=

119899

sum119896=0


119904119886(119899 119896) (119905 + 119886)

119896

(48)



119904119886(119899 119896) = (minus1)

119899minus119896


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902) (49)


119886(119899 119896)| as

1003816100381610038161003816119904119886 (119899 119896)1003816100381610038161003816 = (minus1)

119899minus119896

119904119886(119899 119896)


119899minus119896le119899minus1

119899minus119896

prod119902=1

(119887 + 119895119902)

(50)



lt 1199062


infin

sum119894=1

1

119906119894

= infin

infin

sum119894=1

1

1199062119894

lt infin (51)




which sum119899119896


119870119899= 119899 minus [

119899

sum119894=1

1

119906119894

minus

119899

sum119894=1

1

1199062119894

(1 +1

119906119894

)

minus1

+ 119900 (1)] (52)


1

1+1

2+ sdot sdot sdot +

1

119899= log 119899 + 120574 + 119900 (1) (53)



119886(119899 119896)|

119899

119896=0



119899


mation

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (54)




Denoting by 119896119886119899



1= 119887 + 0 119906

2= 119887 + 1 119906


119896119886119899

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894)2(1 +

1

119887 + 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894minus

119899minus1

sum119894=0

1

(119887 + 119894) (119887 + 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119887 + 119894 + 1+ 119900 (1)]

(55)


119899minus1

sum119894=0

1

119887 + 119894 + 1= log (119887 + 119899) minus log 119887 (56)

From this we get

119896119886119899

= [log(119887 + 119899

119887) + 119900 (1)] (57)



119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (58)



1)(119886 minus


119899minus119896)where 119895


1 By Theorem 10 119904119886(119899 119896) = sum

119899119899minus119896 Thus with 119906

1= 119886 119906

2=

119886 minus 1 119906119899minus1

= 119886 minus 119899 + 1 we have

119896119886119899

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894)2(1 +

1

119886 minus 119894)minus1

+ 119900 (1)]

= [

119899minus1

sum119894=0

1


119899minus1

sum119894=0

1

(119886 minus 119894) (119886 minus 119894 + 1)+ 119900 (1)]

= [

119899minus1

sum119894=0

1

119886 minus 119894 + 1+ 119900 (1)]

(59)


119899119896 0 1 2 30 11 12 2 13 6 6 14 24 35 10 15 120 225 85 156 720 1624 735 1757 5040 13132 6769 19608 40320 109584 118124 672849 362880 1026576 1172700 72368010 3628800 10628640 12753576 8409500


119896119886119899

= [log( 119886 + 1

119886 minus 119899 + 1) + 119900 (1)] (60)


9


119896minus19

= [log(1 + 8

1) + 119900 (1)]

= [log 9 + 119900 (1)]

asymp 2

(61)


10


1198961010

= [log( 10 + 1

10 minus 10 + 1) + 119900 (1)]

= [log 11 + 119900 (1)]

asymp 2

(62)







(63)


Table 2 Values of 11990410(119899 119896)

119899119896 0 1 2 30 11 10 12 90 19 13 720 242 27 14 5040 2414 431 345 30240 19524 5000 6356 151200 127860 44524 81757 604800 662640 305956 772248 1814400 2592720 1580508 5376289 3628800 6999840 5753736 265576410 3628800 10628640 12753576 8409500


119904(119899 119896) 0 1 2 3 4 50 11 0 12 0 1 13 0 2 3 14 0 6 11 6 15 0 24 50 35 10 16 0 120 274 225 85 157 0 720 1764 1624 735 1758 0 540 13068 13132 6769 19609 0 40320 109584 118124 67284 2244910 0 362880 1026576 1172700 723680 269325





2= 2 119906


119896119899= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

1

1198942(1 +

1

119894)minus1

+ 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894minus

119899minus1

sum119894=1

(1

119894minus

1

119894 + 1) + 119900 (1)]

= 1 + [

119899minus1

sum119894=1

1

119894 + 1+ 119900 (1)]

(64)


119899minus1

sum119894=1

1

1 + 119894= log 119899 minus log 1 + 120574 + 119900 (1) (65)

Therefore we have

119896119899= [log 119899 + 120574 + 119900 (1)] (66)



7

119896=0occurs at

1198967= [log 7 + 120574 + 119900 (1)]

= [195 + 05772 + 119900 (1)]

= [253 + 119900 (1)]

asymp 2

(67)


11989610

= [log 10 + 120574 + 119900 (1)]

= [230 + 05772 + 119900 (1)]

= [28772 + 119900 (1)]

asymp 3

(68)




Acknowledgment


References
















1 Introduction















2 New Instructions







x

f(x)

i

(ii) and (iii)1

c













2119888 and



















2minus119890


minus119873

2minus119873

] (1)









2119909 multiplica-








1sdot finite

1+ finite

2= 2 (2)








2120583 in the case






2((maximum



Inputsign exp mant

Drop leading zeros

numZeros + 1


Add

Switch

s e12 m51

lookup output


exp

Add 1

expP1

adjustedE

Switch

bitRot


Add 0x7ff

Add 0x3ff

Switch

Concatenate bit

xor

Add drop carry

moreThan22Zeros

isNot12

Concatenate bit

51

42

42 + 1

44

30

31

52

52

52

52


Add



(c)

(b)

(d)

(e)

(f)

(g)


(h)

(i)

(j)

(k)

leading12Mant

(a)

(p)

(r)

(o)

(l)

(m)

(n)

(q)

(s)

(t)

(u)

(v)

5

sim

sim

0x7ffSubs

Subs 0b0 0




12-bit lookup

12-bit lookup

ge1

e(last 10bits)lt1

sim


0b10 0Subs

Subs 0b0 0

low 10bits4

6

11

11

11

11

12

Lower 10






2119872

recip 3 113 376 0500 2048 3 infin






3 Evaluation





4 Conclusion




(a)


recip (2minus119890

119888)ext (2minus119890



119888)sat

sqrt (2minus119890

119888)ext 21198902

119888 0infin 0 NaN 0 0 0 NaN




(b)


minusinfin 119888 119888 0 0

infin 119888 0 119888 0

NaN 119888 119888 119888 0

(c)



minusinfin119891

minusinfin119891

infin infin119891

infin119891

infin119891

infin119891

NaN0

NaN119891

NaN119891

NaN119891

NaN119891

NaN1

2119891

2119891

2119891

2119891

NaN2

1radic2119891

1radic2119891

1radic2119891

1radic2119891

NaN3

0119891

0119891

0119891

0119891

recipdiv

sqrtrsqrtcbrt

rcbrtqdrt

rqdrtlog2

loglog1pexp2

expexpm1

atan2atanasinacos

sincostan

0 5 10 15 20 25 30







Acknowledgments


References

























1 Introduction








[119865 (119905)]119896

= sum119899gt0

119865 (119899 119896) 119905119899

(1)





119860Δ

(119899 119896) = 120572119896

119865Δ

(119899 119896) (2)



119860Δ

(119899 119896) = 120572119899

119865Δ

(119899 119896) (3)


119891(119899)119905119899 119877(119905) = sum119899gt0

119903(119899)119905119899


119860Δ

(119899 119896) =

119899

sum119898=119896

119865Δ

(119899119898)119866Δ

(119898 119896) (4)


119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899



119886(119899)119905119899

are

119886 (119899) =

119899

sum119896=1

119865Δ

(119899 119896) 119903 (119896)

119886 (0) = 119903 (0)

(5)




119891(119899)119905119899 119877(119905) = sum119899⩾0

119903(119899)119905119899


119899ge0119886(119899)119905119899 are

119886 (119899) =

119899

sum119896=0

119891 (119896) 119903 (119899 minus 119896) (6)




119910119899(119909) =

119899

sum119896=0

(119899 + 119896)

(119899 minus 119896)119896(119909

2)119896

(7)


119901119899(119909) = 119909

119899

119910119899minus1

(1

119909) (8)


119901119899(119909) =

119899

sum119896=1

(2119899 minus 119896 minus 1)

(119896 minus 1) (119899 minus 119896)(1

2)119899minus119896

119909119896 (9)


sum119899=0

119901119899(119909)

119899119905119899


(10)




119862Δ

(119899 119896) =119896

119899(2119899 minus 119896 minus 1

119899 minus 1) (11)



2)

119861 (119905) = 119890119905

=

infin

sum119899=0

1

119899119905119899

(12)


119860Δ

(119899 119896) = (2119909)119896

(1

2)119899

119862Δ

(119899 119896) = 2119896minus119899

119909119896119896

119899(2119899 minus 119896 minus 1

119899 minus 1)

(13)


119899=0119887(119899)119905119899

are equal to

119887 (119899) =1

119899 (14)


119901119899(119909) = 119899

119899

sum119896=1

119860Δ

(119899 119896) 119887 (119896) =

119899

sum119896=1

(2119899 minus 119896 minus 1)


119909119896

(15)




119898119899+1

(119909 120573 119888) =(119888 minus 1)

119899

119888119899

sdot ((119909 minus 119887119899)119898119899(119909 120573 119888) minus 120582

119899119898119899minus1

(119909 120573 119888))

(16)

where

1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) = 119909 minus 119887

0

119887119899=

(1 + 119888) 119899 + 120573119888

1 minus 119888

120582119899=

119888119899 (119899 + 120573 minus 1)

(1 minus 119888)2

(17)


1198980(119909 120573 119888) = 1

1198981(119909 120573 119888) =

119909 (119888 minus 1) + 120573119888

119888

1198982(119909 120573 119888)

=1199092 (119888 minus 1)

2

+ 119909 ((2120573 + 1) 1198882 minus 2120573119888 minus 1) + (120573 + 1) 1205731198882

1198882

(18)


infin

sum119899=0

119898119899(119909 120573 119888)

119899119905119899

= (1 minus119905

119888)119909


(19)





119860 (119905) = (1 minus119905

119888)119909

=

infin

sum119899=0

(119909

119899) (minus1)

119899

119888minus119899

119905119899


=

infin

sum119899=0


119899) (minus1)

119899

119905119899

(20)


suminfin

119899=0119886(119899)119905119899 and 119861(119905) = sum

infin


follows

119886 (119899) = (119909

119899) (minus1)

119899

119888minus119899

119887 (119899) = (minus119909 minus 120573

119899) (minus1)

119899

(21)


119898119899(119909 120573 119888) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896)

= (minus1)119899

119899

119899

sum119896=0

(119909

119896)(


119899 minus 119896) 119888minus119896

(22)



119872119899+1

(119909 120575 120578) = (119909 minus 119887119899)119872119899(119909 120575 120578) minus 120582

119899119872119899minus1

(119909 120575 120578)

(23)

where1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 119887

0

119887119899= (2119899 + 120578) 120575

120582119899= (1205752

+ 1) 119899 (119899 + 120578 minus 1)

(24)


1198720(119909 120575 120578) = 1

1198721(119909 120575 120578) = 119909 minus 120575120578

1198722(119909 120575 120578) = 119909

2

minus 1199092120575 (1 + 120578) + 120578 ((120578 + 1) 1205752

minus 1)

(25)


infin

sum119899=0

119872119899(119909 120575 120578)

119899119905119899

= ((1 + 120575119905)2

+ 1199052

)minus1205782


1 + 120575119905))

(26)




119860 (119905) = ((1 + 120575119905)2

+ 1199052

)minus1205782

= (1 + 2120575119905 + (1205752

+ 1) 1199052

)minus1205782

119861 (119905) = exp (119909tanminus1 ( 119905

1 + 120575119905))

(27)


1(1198602(119905)) and we expand 119860


theorem

1198601(119905) = (1 + 119905)

minus1205782

=

infin

sum119899=0

(minus120578

2119899

) 119905119899

1198602(119905) = 2120575119905 + (120575

2

+ 1) 1199052

(28)


infin

119899=01198861(119899)119905119899

are

1198861(119899) = (

minus120578

2119899

) (29)


1198862119896minus119899

119887119899minus119896

(119896

119899 minus 119896) (30)


119860Δ

2(119899 119896) = (2120575)

2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896) (31)


infin

119899=0119886(119899)119905119899

119886 (0) = 1198861(0)

119886 (119899) =

119899

sum119896=1

119860Δ

2(119899 119896) 119886

1(119896)

(32)


119886 (0) = 1

119886 (119899) =

119899

sum119896=1

(2120575)2119896minus119899

(1205752

+ 1)119899minus119896

(119896

119899 minus 119896)(

minus120578

2119896

) (33)


1(1198612(119905)) where

1198611(119905) = 119890

119905

=

infin

sum119899=0

1

119899119905119899

1198612(119905) = 119909tanminus1 ( 119905

1 + 120575119905)

(34)



functions 1199091198621(1198622(119905)) where

1198621(119905) = tanminus1 (119905)

1198622(119905) =

119905

1 + 120575119905

(35)


((minus1)(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896] (36)

where [119895



119862Δ

1(119899 119896) = ((minus1)

(3119899+119896)2

+ (minus1)(119899minus119896)2

)119896

2119896+1

sdot

119899

sum119895=119896

2119895

119895(119899 minus 1

119895 minus 1)[

119895

119896]

(37)


(119899 minus 1

119896 minus 1) 119886119896

119887119899minus119896

(38)


to

119862Δ

2(119899 119896) = (

119899 minus 1

119896 minus 1) (minus120575)

119899minus119896

(39)


2(119905)

119861Δ

2(119899 119896) = 119909

119896

119899

sum119898=119896

119862Δ

2(119899119898) 119862

Δ

1(119898 119896) (40)


infin

119899=01198871(119899)119905119899

are defined by

1198871(119899) =

1

119899 (41)


infin

119899=0119887(119899)119905119899

119887 (0) = 1198871(0)

119887 (119899) =

119899

sum119896=1

119861Δ

2(119899 119896) 119887

1(119896)

(42)


119887 (0) = 1

119887 (119899) =

119899

sum119896=1

119909119896

119899

119899

sum119898=119896

(119899

119898)120575119899minus119898

(1 + (minus1)

119898+119896

(minus1)(119898+119896)2minus119899

)

sdot

119898

sum119895=119896


(119895 minus 1)(119898

119895)[

119895

119896]

(43)


119872119899(119909 120575 120578) = 119899

119899

sum119896=0

119886 (119896) 119887 (119899 minus 119896) (44)



Acknowledgment


References





































119892 (119911) =1

119911+ 1198870+

infin

sum119899=1

119887119899119911119899 (1)




1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911) (2)






119899| le

(119861 minus 119860)(119899 + 1) for 119892 isin Σ[119860 119861]



ℎ (119908) = 119892minus1

(119908) =1

119908+

infin

sum119899=0

119861119899119908119899

=1


1

119899119870119899

119899+1(1198870 1198871 119887

119899) 119908119899

(3)

where 119908 isin D

119870119899

119899+1(1198870 1198871 119887

119899)

= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)


3119887119899minus4

0(1198874+ 311988711198872)

+ sum119895ge5

119887119899minus119895

0119881119895

(4)


variables 1198871 1198872 119887

119899





119899|



119899| and




119899=1119901119899119911119899 be so that 119877119890(119901(119911)) gt 0


100381610038161003816100381610038161199012+ 1205721199012

1

10038161003816100381610038161003816le 2 + 120572

1003816100381610038161003816119901110038161003816100381610038162

(5)



119899=1119888119899119911119899


100381610038161003816100381610038161198882+ 1205741198882

1

10038161003816100381610038161003816le 1 + (120574 minus 1)

1003816100381610038161003816119888110038161003816100381610038162

(6)

Proof Write

119901 (119911) =[1 + 120593 (119911)]

[1 minus 120593 (119911)] (7)

where 119901(119911) = 1+suminfin



1= 21198881and 119901

2=

2(1198882+ 11988821)


2= 2(1198882+ 11988821) in (5) we

obtain100381610038161003816100381610038162 (1198882+ 1198882

1) + 120572 (2119888

1)210038161003816100381610038161003816le 2 + 120572

10038161003816100381610038162119888110038161003816100381610038162 (8)

or100381610038161003816100381610038161198882+ (1 + 2120572) 119888

2

1

10038161003816100381610038161003816le 1 + 2120572

1003816100381610038161003816119888110038161003816100381610038162

(9)




(119894)10038161003816100381610038161198870

1003816100381610038161003816 le

119861 minus 119860


119861 minus 119860 119900119905ℎ119890119903119908119894119904119890

(119894119894)10038161003816100381610038161198871

1003816100381610038161003816

le

119861 minus 119860


2 (119861 minus 119860)|1198870|2

119894119891 0 le 119860 le 1 minus 2119861

119861 minus 119860

2 119900119905ℎ119890119903119908119894119904119890

(119894119894119894)100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2

(119894V) 10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 119894119891 119887

119896= 0 119891119900119903 0 le 119896 le 119899 minus 1

(10)


1199111198921015840 (119911)

119892 (119911)= minus1 minus

infin

sum119899=0

119865119899+1

(1198870 1198871 1198872 119887

119899) 119911119899+1

(11)

where 119865119899+1

(1198870 1198871 1198872 119887




5=



119865119899+1

(1198870 1198871 119887

119899)


119899+1=119899+1

119860 (1198941 1198942 119894

119899+1) 1198871198941

01198871198942

1sdot sdot sdot 119887119894119899+1

119899

(12)

where

119860 (1198941 1198942 119894

119899+1)

= (minus1)(119899+1)+2119894

1+sdotsdotsdot+(119899+2)119894

119899+1

(1198941+ 1198942+ sdot sdot sdot + 119894

119899+1minus 1) (119899 + 1)

(1198941) (1198942) sdot sdot sdot (119894

119899+1)

(13)



119908ℎ1015840 (119908)


infin

sum119899=1

119865119899+1

(1198610 1198611 1198612 119861

119899) 119908119899+1

(14)

where119865119899+1

(1198610 1198611 1198612 119861


119899 + 1 given by

119865119899+1

= minus119899 (119899 minus (119899 + 1))

119899 (119899 minus 2119899)119861119899

0

minus119899 (119899 minus (119899 + 1))


01198611

minus119899 (119899 minus (119899 + 1))


01198612

minus119899 (119899 minus (119899 + 1))


0

sdot (1198613+119899 minus (2119899 minus 3)

21198612


119861119899minus119895

0119870119895

(15)


variables 1198611 1198612 119861

119899minus1and

1198610= minus1198870 119861

119899= minus

1

119899119870119899

119899+1(1198870 1198871 119887

119899) (16)


1199111198921015840

(119911)

119892 (119911)= minus

1 + 119860120593 (119911)

1 + 119861120593 (119911)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198881 1198882 119888

119899 119861) 119911119899

(17)

119908ℎ1015840 (119908)

ℎ (119908)= minus

1 + 119860120595 (119908)

1 + 119861120595 (119908)

= minus1 +

infin

sum119899=1

(119860 minus 119861)119870minus1

119899(1198891 1198892 119889

119899 119861) 119908

119899

(18)


119899=1119888119899119911119899 and 120595(119908) = sum

infin

119899=1119889119899119908119899 are two



119899(1198961 1198962 119896

119899 119861) are given by

119870119901

119899(1198961 1198962 119896

119899 119861)

=119901

(119901 minus 119899)119899119896119899

1119861119899minus1

+119901


11198962119861119899minus2

+119901


11198963119861119899minus3

+119901


1

sdot [1198964119861119899minus4

+119901 minus 119899 + 3

21198962

2119861]

+119901


1

sdot [1198965119861119899minus5

+ (119901 minus 119899 + 4) 11989621198963119861]

+ sum119895ge6

119896119899minus119895

1119883119895

(19)


variables 1198962 1198963 119896

119899


minus119865119899+1

(1198870 1198871 1198872 119887

119899) = (119860 minus 119861)119870

minus1

119899+1(1198881 1198882 119888

119899+1 119861)

(20)


minus 119865119899+1

(1198610 1198611 1198612 119861

119899)

= (119860 minus 119861)119870minus1

119899+1(1198891 1198892 119889

119899+1 119861)

(21)


1198870= (119860 minus 119861) 119888

1

minus1198870= (119860 minus 119861) 119889

1

(22)

21198871minus 1198872

0= (119861 minus 119860) (119888

2

1119861 minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (119889

2

1119861 minus 1198892)

(23)




0and then adding

them gives

21198872

0= (119861 minus 119860) (119888

2+ 1198892minus 1198611198882

1minus 1198611198892

1) (24)



21198872

0= (119861 minus 119860) (119888

2+ 1198892minus

2119861

(119861 minus 119860)21198872

0) (25)


sides we obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(119861 minus 119860)

2

(1198882+ 1198892)

2 (2119861 minus 119860)le

119861 minus 119860

radic2119861 minus 119860 (26)



119861 minus 119860



we obtain

41198871= (119861 minus 119860) [(119889

2minus 1198882) minus (119889

2

1minus 1198882

1) 119861] = (119861 minus 119860) (119889

2minus 1198882)

(28)


100381610038161003816100381611988711003816100381610038161003816 le

119861 minus 119860

2 (29)


21198871= (119861 minus 119860) (119889

2+ 1198601198892

1minus 2119861119889

2

1) (30)


100381610038161003816100381611988711003816100381610038161003816 le

1

2(119861 minus 119860) (

100381610038161003816100381610038161198892+ 1198601198892

1

10038161003816100381610038161003816+ 2119861

1003816100381610038161003816119889110038161003816100381610038162

)

le1

2(119861 minus 119860) (1 + (119860 minus 1)

1003816100381610038161003816119889110038161003816100381610038162

+ 211986110038161003816100381610038161198891

10038161003816100381610038162

)

=119861 minus 119860


2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

(31)




2 (119861 minus 119860)

1003816100381610038161003816119887010038161003816100381610038162

lt119861 minus 119860

2 (32)


21198871minus 1198872

0= (119861 minus 119860) (

119861

(119861 minus 119860)21198872

0minus 1198882)

21198871+ 1198872

0= (119860 minus 119861) (

119861

(119860 minus 119861)21198872

0minus 1198892)

(33)


100381610038161003816100381610038161003816100381610038161198871plusmn

2119861 minus 119860

2 (119861 minus 119860)1198872

0

10038161003816100381610038161003816100381610038161003816le119861 minus 119860

2 (34)


(119899 + 1) 119887119899= (119860 minus 119861) 119888

119899+1 (35)


obtain

10038161003816100381610038161198871198991003816100381610038161003816 le

119861 minus 119860

119899 + 1 (36)







0|2 given by






References



























1 Introductions


119891 (119911) = 119911 +

infin

sum120581=2

119886120581119911120581

(1)


119867[119886 119899] = 119901 isin 119867 119901 (119911) = 119886 + 119886119899119911119899

+119886119899+1

119911119899+1


(2)


119865 (119911) = 119911120573

+

infin

sum120581=2

120582120581119886120581119911120573+120581minus1

120573 isin N

120582120581= 120573120581minus1

119911 =1

120573 120573 |119911| lt 1

(3)


(119891 lowast 119892) (119911) =

infin

sum119896=0

119886119896119887119896119911119896

(4)


119896=0119886119896119911119896 and 119892(119911) = sum

infin

119896=0119887119896119911119896 are analytic in

119880

Let 120595(119911) = 119911120573 + sum

infin



120595 (119911) lowast119911120573

1 minus 120573119911 119911 =

1

120573 120573 isin N 120573 |119911| lt 1 (5)



119876(120485) (120572 120591 120574 120573)

= 119865 (119911) isin 119867 (120573) R119890

times120572[119865(119911)]

(120485)

[119911120573](120485)

+120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

gt 120574 119911 isin 119880

(6)

where

119911 = 0 120573 gt 120485 120485 isin N cup 0 also 120572 + 120591 = 120574 (7)







119891 (119911) =1

120572 + 120591int|119909|


times

infin

sum120581=0

(120572 + 120591) 119909120581

119911120581+1

(120581 + 1) 120591 + 120572] 119889120583 (119909)

(8)




Θ120572120573

(119911) = (120573int119911

0

119905120573minus120572minus1

[119891 (119905)]120572

119892 (119905) 119889119905)

1120573

(10)






119901 (119911) = int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909) (|119911| lt 1) (11)



119892 (119911) +1199111198921015840 (119911)

119888≺ ℎ (119911) (12)

then

119892 (119911) ≺ 119902 (119911) =119888119911minus119888120581

120581int119911

0

119905(119888minus120581)120581

ℎ (119905) 119889119905 ≺ ℎ (119911) (13)



1199111199011015840 (119911)

119901 (119911)≺ ℎ (119911) (14)

then

119901 (119911) ≺ 119902 (119911) = 119886 exp [1

119899int119911

0

ℎ (120585) 120585minus1

119889120585] (15)




(ii) |1199011015840(119911)| le 2R119890119901(119911)(1 minus 1199032)


100381610038161003816100381610038161003816100381610038161003816

1199111199011015840 (119911)

119901 (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 119901 (119911)

(1 minus 119903)2

(16)




(120485)

(120572120591120574120573)




119911120485

[119865 (119911)](120485)

=

120573

119875120485

(120572 + 120591)int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120573 minus 120485) (120572 + 120591) 119909120581119911120573+120581

(120572 + 120591) (120573 minus 120485) + 120581120591] 119889120583 (119909)

(17)

where 119899119875119903




119901 (119911)

=

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

isin 119867 [1 119899]

(18)

ByTheorem 3

(120591[119865 (119911)](120485+1)

[119911120573](120485+1)

) + (120572[119865 (119911)](120485)

[119911120573](120485)

) minus 120574

120572 + 120591 minus 120574

= int|119909|=1

1 + 119909119911

1 minus 119909119911119889120583 (119909)

(19)


[119865 (119911)](120485+1)

[119911120573](120485+1)

+120572[119865 (119911)]

(120485)

120591[119911120573](120485)

=1

120591int|119909|=1

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909119911

1 minus 119909119911119889120583 (119909)

(20)

which yields

119911((120572+120591)(120485minus120573)+120591)120591

int119911

0

[[119865 (120585)]

(120485+1)

[120585120573](120485+1)

+120572[119865 (120585)]

(120485)

120591[120585120573](120485)

]


119889120585

=1

120591int|119909|=1

119911((120572+120591)(120485minus120573)+120591)120591

times int119911

0

(120572 + 120591) + (120572 + 120591 minus 2120574) 119909120585

1 minus 119909120585


119889120585 119889120583 (119909)

(21)



shows that 119865(119911) isin 119876(120485)(120572 120591 120574 120573)



119865 (119911) =1

120572 + 120591int|119909|=1

[ (2120574 minus 120572 minus 120591) 119911120573

+ 2 (120572 + 120591 minus 120574)

times

infin

sum120581=0

(120572 + 120591) 119909120581119911120573+120581

120572120573 + 120591 (120581 + 120573)] 119889120583 (119909)

(22)


1




10038161003816100381610038161198861205811003816100381610038161003816 le 119867 (120572 120591 120574 120573) (23)

where

119867(120572 120591 120574 120573) =2 (120572 + 120591 minus 120574) (120573 minus 120485) [

120573

119875120485]

120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(24)


119911120485

[119865119909(119911)](120485)

120573119875120485

= 119911120573

+

infin

sum120581=2


(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)(|119909| = 1)

(25)




119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [[120581 minus 1] 119903

120581minus1]

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

le2119903R119890 Ψ (119911)

(1 minus 119903)2

(26)


119866 (119911) equiv120591[119865(119911)]

(120485+1)

[119911120573](120485+1)

+120572[119865(119911)]

(120485)

[119911120573](120485)

minus 120574 (27)

and then119911Ψ1015840 (119911)

Ψ (119911)=

suminfin

120581=2120599 (120572 120591 120574 120573) [120581 minus 1] 119886

120581119911120581minus1

1 + suminfin

120581=2120599 (120572 120591 120574 120573) 119886

120581119911120581minus1

(28)

where

120599 (120572 120591 120574 120573) =120582120581[ [120573+120581minus1]119875

120485] [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591 minus 120574) [120573

119875(120485)

]

(29)

From (23) and (28) we got100381610038161003816100381610038161003816100381610038161003816

119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

suminfin

120581=22 (120573 minus 120485) [120581 minus 1] 119903

120581minus1

1 + suminfin

120581=22 (120573 minus 120485) 119903120581minus1

(120581 ge 2) (30)


119911Ψ1015840 (119911)

Ψ (119911)

100381610038161003816100381610038161003816100381610038161003816le

2119903R119890 Ψ (119911)

(1 minus 119903)2

(31)

Sinceinfin

sum120581=2


lt 2119903R119890 Ψ (119911)

1 +

infin

sum120581=2

2 (120573 minus 120485) 119903120581minus1

gt (1 minus 119903)2

(32)




(120485)

(120572 120591 120574 120573)



1199111198661015840 (119911)

119866 (119911) ≺ ℎ (119911) (33)

then

119866 (119911) ≺ 119902 (119911) = (120572 + 120591 minus 120574) exp [119899minus1

int119911

0

ℎ (120585) 120585minus1

119889120585 ] (34)



119866 (119911) = (120572 + 120591 minus 120574) +

infin

sum120581=2

120582120581[ [120573+120581minus1]119875

120485+1] 119886120581119911120581minus1

120573119875120485+1

(35)


1199111198661015840

(119911)

119866 (119911) ≺ ℎ (119911) (36)




1199111198661015840 (119911)

119866 (119911) ≺

radic119911

1 minus 119911(37)

then

119866 (119911) ≺ (120572 + 120591 minus 120574) 1 + radic119911

1 minus radic119911

1119899

120572 + 120591 = 120574 (38)




120573 gt 120485 + 1 with 120572 + 120591 = 0 If

119866 (119911) ≺ (120572 + 120591)1 + 119860119911

1 + 119861119911 (39)

then[119865(119911)]

(120485)

[119911120573](120485)

≺(120572 + 120591) (120573 minus 120485)

120591120581119911(120572+120591)(120485minus120573)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(40)


1

(120572 + 120591)119866 (119911) = 1 +

infin

sum120581=2

119862120581119886120581119911120581minus1

(41)

where

119862120581=

120582120581[ (120573+120581minus1)119875

120485+1] sdot [(120572 + 120591) (120573 minus 120485) + 120591 (120581 minus 1)]

(120572 + 120591) [120573

119875120485+1

] (42)

Let 119892(119911) = [119865(119911)](120485)

[119911120573](120485) then 119892(119911) = 1 + sum

infin

120581=2120603120581119886120581119911120581minus1

where 120603120581= 120582120581[(120573+120581minus1)

119875120485+1

][(120573)

119875120485] and from (39) we have

119892 (119911) +120591

(120572 + 120591) (120573 minus 120485)1199111198921015840

(119911) ≺ (1 + 119860119911

1 + 119861119911) = ℎ (119911) (43)


119892 (119911) ≺(120572 + 120591) (120573 minus 120485)

120591120581119911minus(120572+120591)(120573minus120485)120591120581

times int119911

0


(1 + 119860120585

1 + 119861120585)119889120585

(44)


Corollary 16 Let

119876(0)

(120572 120591 0 120573) =120572 (119865 (119911))

119911120573+

120591[119865 (119911)]1015840

[119911120573]1015840

(45)

If

119876(0)

(120572 120591 0 120573) ≺ (120572 + 120591) (1 + 119860119911

1 + 119861119911) (46)

then

119865 (119911)

119911120573≺

120572120573 + 120591 (120573 minus 120581)

120591120581119911minus120573(120572+120591)120591120581

times int119911

0

120585(120573(120572+120591)minus120591120581)120591120581

(1 + 119860120585

1 + 119861120585)119889120585

(47)






Acknowledgments



References





















1 Introduction


119891 (119911) =119892 (119911)

ℎ (119911) (1)






R ℎ (119911) gt 0 (119911 isin 119880lowast

) (2)


119891 ≺ 119892 in 119880 or 119891 (119911) ≺ 119892 (119911) (119911 isin 119880lowast

) (3)


|119908 (119911)| le |119911| 119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880lowast

) (4)


119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0)

119891 (119880) sube 119892 (119880) (119911 isin 119880lowast

) (5)





2 Preliminaries


119891 (119911) =1

119911+

infin

sum119899=1

119886119899119911119899

(6)




119891119895(119911) =

1

119911+

infin

sum119899=1

119886119899119895119911119899

(7)


and 1198912(119911) by

(1198911lowast 1198912) =

1

119911+

infin

sum119899=1

11988611989911198861198992119911119899

(8)


119902120582120583

(119911) =1

119911+

infin

sum119899=1

(120582

119899 + 1 + 120582)

120583

119911119899

(120582 gt 0 120583 ge 0) (9)



119871120582120583119891 (119911) = (119891 (119911) lowast 119902

120582120583(119911))

=1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(10)


119905119886119892 and 119871


tively by

119871120582120583119892 (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198871198991003816100381610038161003816 119911119899


119871120582120583ℎ (119911) =

1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198881198991003816100381610038161003816 119911119899


ℎ (119911) isin Σ)

(11)


119871120582120583

(119891 lowast 119892) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198871198991003816100381610038161003816 119911119899

(119911 isin 119880lowast)

119871120582120583

(119891 lowast ℎ) (119911) =1

119911+

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991198881198991003816100381610038161003816 119911119899


)

(12)






property

120588119871120582120583

(119891 lowast 119892) (119911)

119871120582120583

(119891 lowast ℎ) (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911 (119911 isin 119880

lowast

) (13)


and 119871(120582 120583)(119891 lowast ℎ)(119911) = 0


120601 (120572 120573 119911) =1

119911+

infin

sum119899=0

(120572)119899+1

(120573)119899+1

119911119899

(14)



120601 (120572 120573 119911) =1

119911 21198651(1 120572 120573 119911) (15)

where

21198651(119887 120572 120573 119911) =

infin

sum119899=0

(119887)119899(120572)119899

(120573)119899

119911119899

119899(16)


120582120583(119911)


119871 (120572 120573 120582 120583) 119891 (119911) = (119891 (119911) lowast 120601 (120572 120573 119911) lowast 119902120582120583

(119911))

=1

119911+

infin

sum119899=1

(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

(17)




119911(119871 (120572 120573 120582 120583) 119891 (119911))1015840

= 120572119871 (120572 + 1 120573 120582 120583) 119891 (119911)

minus (120572 + 1) 119871 (120572 120573 120582 120583) 119891 (119911) (18)


120588119871 (120572 + 1 120573 120582 120583) 119891 (119911)

119871 (120572 120573 120582 120583) 119891 (119911)≺ 120588 minus

(119860 minus 119861) 119911

1 + 119861119911

(0 le 119861 lt 119860 le 1 120588 gt 0)

(19)



relation (19) above






infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861)

minus10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(20)


119891119899(119911) =

1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

(21)



1003816100381610038161003816100381610038161003816100381610038161003816

120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899) 119911119899+1

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899119911119899+1

)

minus11003816100381610038161003816100381610038161003816100381610038161003816

le 120588

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

119886119899(119887119899minus 119888119899)10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816

times ((119860 minus 119861)

minus

infin

sum119899=1

(120582

119899 + 120582 + 1)

120583


119899

10038161003816100381610038161003816119911119899+1

10038161003816100381610038161003816)

minus1

le 1

(22)





100381610038161003816100381610038161003816100381610038161003816lt 1

(119911 isin 119880lowast)

(23)


Σ120583




120582(120588 119860 119861) then

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119899 ge 1

(24)


(21)


120582(120588 119860 119861)



1

119903minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903

le1003816100381610038161003816119891 (119911)

1003816100381610038161003816

le1

119903+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119903


1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

le100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(25)



inequalityinfin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 le

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(26)


1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

le1

119903+ 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

119903+ 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

1003816100381610038161003816119891 (119911)1003816100381610038161003816 =

1

|119911|minus

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899

ge1

119903minus 119903

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

ge1

119903minus 119903

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(27)


sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(119860 minus 119861) (2 + 120582)

120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(28)

Hence

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2+

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

le1

1199032+

infin

sum119899=1

10038161003816100381610038161198861198991003816100381610038161003816

le1

1199032+

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

100381610038161003816100381610038161198911015840

(119911)10038161003816100381610038161003816=

1

|119911|2minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

ge1

1199032minus

infin

sum119899=1

1198991003816100381610038161003816119886119899

1003816100381610038161003816

ge1

1199032minus

(119860 minus 119861) (2 + 120582)120583

120582120583 (120588100381610038161003816100381611988711003816100381610038161003816 (1 + 119861) minus

100381610038161003816100381611988811003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

(29)



120582(120588 119860 119861) which




disk |119911| lt 1199031 where

1199031= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(30)



119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (31)

For |119911| lt 1199031 we have

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891(119911))1015840

119891 (119911)+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 1 + 120582))

120583

119886119899119911119899

1119911 + suminfin

119899=1(120582(119899 + 1 + 120582))

120583

119886119899119911119899

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

suminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583

119886119899119911119899+1

100381610038161003816100381610038161003816100381610038161003816

lesuminfin

119899=1(119899 + 1) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

1 + suminfin

119899=1(120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(32)


sum119899=1

(119899 + 1) (120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1


infin

sum119899=1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

)

(33)

or

suminfin

119899=1(119899 + 2 minus 120575) (120582(119899 + 120582 + 1))

120583 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899+1

(1 minus 120575)le 1 (34)


(119899 + 2 minus 120575) (120582(119899 + 1 + 120582))120583

|119911|119899+1

(1 minus 120575)

le (120582

119899 + 120582 + 1)

120583

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899 ge 1

(35)



|119911| le (1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) +

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


119899+1

(36)




disk |119911| lt 1199032 where

1199032= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(37)



1003816100381610038161003816100381610038161003816100381610038161003816

119911(119891 (119911))10158401015840

(119891 (119911))1015840+ 2

1003816100381610038161003816100381610038161003816100381610038161003816

le 1 minus 120575 (38)






infin

sum119899=1

(12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

times(120572)119899+1

(120573)119899+1

(120582

119899 + 120582 + 1)

120583

10038161003816100381610038161198861198991003816100381610038161003816 le 119860 minus 119861

(39)


119891119899(119911)

=1

119911

+

infin

sum119899=1

(119860 minus 119861) (119899 + 120582 + 1)120583

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))

119911119899

119899 ge 1

(40)




120582(120588 120572 120573 119860 119861) then1003816100381610038161003816119886119899

1003816100381610038161003816

le(119860 minus 119861) (119899 + 120582 + 1)

120583

(120573)119899+1

120582120583 (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861)) (120572)

119899+1

119899 ge 1

(41)


(40)




1199033= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(42)





1199034= inf119899ge1

(1 minus 120575)

times (12058810038161003816100381610038161198871198991003816100381610038161003816 (1 + 119861) minus

10038161003816100381610038161198881198991003816100381610038161003816 (120588 (1 + 119861) + 119860 minus 119861))


1(119899+1)

(43)






Acknowledgment


References






















1 Introduction


Φ (119911 119904 119886) =

infin

sum119899=0

119911119899

(119899 + 119886)119904


0 119904 isin C when |119911| lt 1

R (119904) gt 1 when |119911| = 1)

(1)


119904(120585) defined by

120577 (119904) =

infin

sum119899=1

1

119899119904= Φ (1 119904 1) = 120577 (119904 1) (R (119904) gt 1)

120577 (119904 119886) =

infin

sum119899=0

1

(119899 + 119886)119904


0)

ℓ119904(120585) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 1)119904= Φ (e2120587119894120585 119904 1) (R (119904) gt 1 120585 isin R)

(2)


119904(119911)

Li119904(119911) =

infin

sum119899=1

119911119899

119899119904= 119911Φ (119911 119904 1)


(3)


120601 (120585 119886 119904) =

infin

sum119899=0

e2119899120587119894120585

(119899 + 119886)119904= Φ (e2120587119894120585 119904 119886)



when 120585 isin Z)

(4)




Φ (119911 119904 119886) =1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

1 minus 119911eminus119905d119905

(R (119886) gt 0R (119904) gt 0 when |119911| ≦ 1 (119911 = 1) R (119904) gt 1

when 119911 = 1) (5)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) =

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

119911119899

(119899 + 119886)119904

(119901 119902 isin N0 120582

119895isin C (119895 = 1 119901)

119886 120583119895isin C Z

minus

0(119895 = 1 119902)

120588119895 120590

119896isin R

+

(119895 = 1 119901 119896 = 1 119902)





lowast

)

(6)

with

nablalowast

= (

119901

prod119895=1

120588minus120588119895

119895) sdot (

119902

prod119895=1

120590120590119895

119895)

Δ =

119902

sum119895=1

120590119895minus

119901

sum119895=1

120588119895

Ξ = 119904 +

119902

sum119895=1

120583119895minus

119901

sum119895=1

120582119895+119901 minus 119902

2

(7)



(120582)120581=Γ (120582 + 120581)

Γ (120582)

=


1 (120581 = 0 120582 isin C 0)

(8)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0)

(9)



119898or

119897Ψ119898(119897 119898 isin N

0) involved


119897119865119898(119897 119898 isin N




1198871 119887

119898such that

119886119895isin C (119895 = 1 119897) 119887

119895isin C Z

minus

0(119895 = 1 119898)

(10)


119897Ψlowast

119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

=

infin

sum119899=0

(1198861)1198601119899sdot sdot sdot (119886

119897)119860119897119899

(1198871)1198611119899sdot sdot sdot (119887

119898)119861119898119899

119911119899

119899

=Γ (119887


119898)

Γ (1198861) sdot sdot sdot Γ (119886

119897) 119897Ψ119898[(119886

1 119860

1) (119886

119897 119860

119897)

(1198871 119861

1) (119887

119898 119861

119898) 119911 ]

(119860119895gt 0 (119895 = 1 119897) 119861

119895gt 0 (119895 = 1 119898)

1 +

119898

sum119895=1

119861119895minus

119897

sum119895=1

119860119895≧ 0)

(11)


|119911| lt nabla = (

119897

prod119895=1

119860minus119860119895

119895) sdot (

119898

prod119895=1

119861119861119895

119895) (12)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

Γ (119904)intinfin

0


119905120582)

sdot119901Ψlowast

119902[(120582

1 120588

1) (120582

119901 120588

119901)

(1205831 120590

1) (120583

119902 120590

119902) 119911eminus119905] d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0)

(13)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 0 120582)

= Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886)

= e119887Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 0)

(14)


119901 minus 1 = 119902 = 0 (1205821= 120583 120588

1= 1) (15)


Θ120582

120583(119911 119904 119886 119887)

=1

Γ (119904)intinfin

0


119905120582) (1 minus 119911eminus119905)

minus120583

d119905

(min R (119886) R (119904) gt 0 R (119887) ≧ 0 120582 ≧ 0 120583 isin C)

(16)


R (119904) gt 0 when 119887 = 0 |119911| ≦ 1 (119911 = 1) (17)

or

R (119904 minus 120583) gt 0 when 119887 = 0 119911 = 1 (18)




Θ120582


Θ120582

120583(119911 119904 119886 0) = Φ

lowast

120583(119911 119904 119886) =

1

Γ (119904)intinfin

0

119905119904minus1eminus119886119905

(1 minus 119911eminus119905)120583d119905

(R (119886) gt 0 R (119904) gt 0 when |119911| ≦ 1 (119911 = 1)

R (119904 minus 120583) gt 0 when 119911 = 1)

(19)



[4] as follows

Φlowast

120583(119911 119904 119886) =

infin

sum119899=0

(120583)119899

(119886 + 119899)119904

119911119899

119899 (20)


120583(119911 119904 119886 119887) that is


Θ1

1(1 119904 119886 119887)

= 120577119887(119904 119886)

=1

Γ (119904)intinfin

0

119905119904minus1


119905) d119905

(21)




(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899

119899(120582 gt 0)

(22)



119867119898119899

pq (119911)

= 119867119898119899

pq [119911 |(119886

1 119860

1) (119886p 119860p)

(1198871 119861

1) (119887q 119861q)

]

=1

2120587119894intL

Ξ (119904) 119911minus119904d119904 (119911 isin C 0

1003816100381610038161003816arg (119911)1003816100381610038161003816 lt 120587)

(23)

where

Ξ (119904) =prod

119898

119895=1Γ (119887

119895+ 119861

119895119904) sdot prod

119899

119895=1Γ (1 minus 119886

119895minus 119860

119895119904)

prodp

119895=119899+1Γ (119886

119895+ 119860

119895119904) sdot prod

q

119895=119898+1Γ (1 minus 119887

119895minus 119861

119895119904)

(24)


119895gt 0 (119895 =

1 p) 119861119895gt 0 (119895 = 1 q) 119886

119895isin C (119895 = 1 p)



Γ (119887119895+ 119861

119895119904)

119898

119895=1

(25)


Γ (1 minus 119886119895+ 119860

119895119904)

119899

119895=1

(26)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902





1 minus 119905)

=

infin

sum119899=0

119871(120572)

119899(119909) 119905

119899

(|119905| lt 1 120572 isin C)

(27)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=eminus119887

Γ (119904)

infin

sum119899=0

119899

sum119896=0

(minus1)119896

(119899

119896) sdot Γ (119904 + 120582 (120572 + 119896 + 1))

sdot 119871(120572)

119899(119887)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 + 120582 (120572 + 119895 + 1) 119886)

(R (119886) gt 0 R (119904 + 120582120572) gt minus120582)

(28)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886) (29)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



D120572

119911119911

119901

119891 (119911) =1

Γ (minus120572)int119911

0

119891 (120585) 120585119901


(R (120572) lt 0 R (119901) gt 1)

(30)


D120572

119911119911

119901

119891 (119911) =d119898

d119911119898D

120572minus119898

119911119911

119901

119891 (119911) (31)



Im(120585)

Branch line for

Branch line for


exp[p(ln(g(120585)))]

C1

C2

C3

C4

Re(120585)

z

a

gminus1(0)





119901


119863120572

119892(119911)[119892 (119911)]

119901

119891 (119911)

=eminus119894120587119901Γ (1 + 120572)4120587 sin (120587119901)

times int119862(119911+119892

minus1(0)+119911minus119892

minus1(0)minus119865(119886)119865(119886))

sdot119891 (120585) [119892 (120585)]

119901

1198921015840 (120585)

[119892 (120585) minus 119892 (119911)]120572+1

d120585

(32)



1cup119862

2cup119862

3cup119862








119892(119911)[119892(119911)]

119901



119863120572

119911119911

119901

=Γ (1 + 119901)

Γ (1 + 119901 minus 120572)119911119901minus120572

(R (119901) gt minus1) (33)



11205881199011205901120590119902)

12058211205821199011205831120583119902


119863120572

119911119911

120573minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899)

1198871120582

| (119904 1) (01

120582)]119863

120572

119911119911

120573minus1119911119899

119899

=Γ (120573)

120582Γ (119904) Γ (120573 minus 120572)

infin

sum119899=0

(120573)119899

(120573 minus 120572)119899

sdot

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

sdot 11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)]

119911119899+120573minus120572minus1

119899

=Γ (120573)


Φ(1205881120588119901112059011205901199021)

12058211205821199011205731205831120583119902120573minus120572

(119911 119904 119886 119887 120582)







119863120572

119892(119911)119891 (119911) = 119863

120572

ℎ(119911)119891 (119911) 119892

1015840(119911)

ℎ1015840(119911)(ℎ(119908) minus ℎ(119911)

119892(119908) minus 119892(119911))

120572+1

100381610038161003816100381610038161003816100381610038161003816119908=119911

(35)




119911119901119874120572

120573119891 (119911)

= 119901(119911119901minus1

)minus120572

sdot119911

119874120572

120573119891 (119911) (119911

119901minus1

)120572

119901minus1

prod119904=1

(1 minus119911

119908eminus(2120587119894119904)119901)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(36)

where

119892(119911)119874120572


Γ (120573)

Γ (120572)[119892 (119911)]

1minus120573

119863120572minus120573

119892(119911)[119892 (119911)]

120572minus1

sdot sdot sdot

(37)



120579 (119911) = (119911 minus 1199110) 119902 (119911) (38)


119870

=0 1 [119888] ([119888] 119905ℎ119890 119897119886119903119892119890119904119905 119894119899119905119890119892119890119903 119899119900119905 119892119903119890119886119905119890119903 119905ℎ119886119899 119888)

(39)


0and let

120596 = exp (2120587119894119886) (40)



sum119896isin119870

119888minus1

120596minus120574119896

119891 (120579minus1

(120579 (119911) 120596119896

))

=

infin


[120579 (119911)]119888119899+120574

Γ (119888119899 + 120574 + 1)

sdot119863119888119899+120574

119911minus119887119891 (119911) 120579

1015840

(119911) (119911minus119911

0

120579 (119911))

119888119899+120574+1

100381610038161003816100381610038161003816100381610038161003816119911=1199110

(1003816100381610038161003816119911minus1199110

1003816100381610038161003816=100381610038161003816100381611991101003816100381610038161003816)

(41)



119891 (119911) = 119888

infin


(119911 minus 1199110)119888119899+120574

Γ(119888119899 + 120574 + 1)119863

119888119899+120574

119911minus119887119891(119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199110

(42)



1)(119911minus119911

2)) where 119911

1and 119911

2are two



120596 = exp(2120587119894119886) (43)


1and 119911


curves

119862 (119905) 119862 (119905) subR 0 lt 119905 ≦ 119903 (44)

be defined by

119862 (119905) = 1198621(119905) cup 119862

2(119905)

= 119911 1003816100381610038161003816120582119905 (1199111 1199112 119911)

1003816100381610038161003816 =1003816100381610038161003816100381610038161003816120582119905(119911

1 119911

21199111+ 119911

2

2)1003816100381610038161003816100381610038161003816

(45)

where

120582119905(119911

1 119911

2 119911) = [119911 minus

1199111+ 119911

2

2+ 119905 (

1199111minus 119911

2

2)]

sdot [119911 minus (1199111+ 119911

2

2) minus 119905 (

1199111minus 119911

2

2)]

(46)


1+ 119911



1199111+ 119911

2

2+ (

1199111minus 119911

2

2) 119905 (47)


2

2minus (

1199111minus 119911

2

2) 119905 (48)

Im(120585)

Branch line for

Branch line for

C1

C2

z1

z2

Re(120585)

z

c =z1 + z2

2

(120585 minus z2)120583120579(120585)

aminus120574minus1

(120585 minus z1)120579(120585)

aminus120574minus1



[(119911 minus 1199111) (119911 minus 119911

2)]

120582


2))))

(49)


plusmn

defined by

119871plusmn=

119911 119911 =1199111+ 119911

2

2plusmn 119905 (

1199111minus 119911

2

2) (0 ≦ 119905 ≦ 1)

119911 119911 =1199111+ 119911

2

2plusmn 119894119905 (

1199111minus 119911

2

2) (119905 lt 0)

(50)



1)(119911minus119911

2) gt 0 (v)


2)119901


+cup 119871


119911minus1199112

(119911 minus 1199112)119901


119870 = 119896 119896 isin N arg(120582119905(119911

1 119911

21199111+ 119911

2

2))

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) +

2120587119896

119886

lt arg(120582119905(119911

1 119911

21199111+ 119911

2

2)) + 2120587

(51)


defined by

120585 =1199111+ 119911

2

2+1199111minus 119911

2


sum119896isin119870

119888minus1120596minus120574119896

1199111minus 119911

2

119891 (120601minus1

(120596119896

120601 (119911)))

times [120601minus1

(120596119896

120601(119911)) minus 1199111]][120601

minus1

(120596119896

120601(119911)) minus 1199112]120583

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

[120601 (119911)]119888119899+120574

(52)


where

120601 (119911) =119911 minus 119911

1

119911 minus 1199112

(53)



2)120583

1199111minus 119911

2

=

infin



sdot119863minus]+119888119899+120574119911minus1199112

(119911 minus 1199112)120583+119888119899+120574minus1

119891(119911)10038161003816100381610038161003816119911=1199111

(119911 minus 119911

1

119911 minus 1199112

)

119888119899+120574

(54)


2) where 119911

1and 119911




sum119896isin119870

119888minus1

120596minus120574119896

[(1199112minus 119911

1+ radicΔ

119896

2)

120572

(1199111minus 119911

2+ radicΔ

119896

2)

120573

sdot 119891 (1199111+ 119911

2+ radicΔ

119896

2)


sdot (1199112minus 119911

1minus radicΔ

119896

2)

120572


2minus radicΔ

119896

2)

120573

119891(1199111+ 119911

2minus radicΔ

119896

2)]

=

infin



eminus119894120587119888(119899+1)[120579 (119911)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911minus1199112


(120579 (119911)

(119911minus1199112) (119911minus119911

1))


1205791015840

(119911)119891 (119911)

1003816100381610038161003816100381610038161003816100381610038161003816119911=1199111

(55)

where

Δ119896= (119911

1minus 119911

2)2

+ 4119881 (120596119896

120579 (119911))

119881 (119911) =

infin

sum119903=1

119863119903minus1

119911[119902 (119911)]

minus119903

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

119911119903

119903

120579 (119911) = (119911 minus 1199111) (119911 minus 119911

2) 119902 ((119911 minus 119911

1) (119911 minus 119911

2))

(56)


(119911 minus 1199111)(119911 minus 119911

2)) and 119911


119891 (119911) = 119888119911minus120573

(119911 minus 1199111)minus120572

times

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot119863minus120572+119888119899+120574

119911119911


(119911 + 119908 minus 1199111)119891(119911)

10038161003816100381610038161003816119911=1199111

(119908=119911)

(57)



119863120572

119911119911

119901+119902

119906 (119911) V (119911)

=

infin


(120572

120574 + 119899)119863

120572minus120574minus119899

119911119911

119901

119906 (119911)119863120574+119899

119911119911

119902V (119911) (58)




119863120572

119911119911

120572+120573minus1

119906 (119911) V (119911)


sdot

infin


119863120572+]+1minus119899119911



(60)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)



Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 (1120582))]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot 119865(119896minus1)



where 120582 gt 0 and 119865(119899)119863


119865(119899)

119863[119886 119887

1 119887

119899 119888 119909

1 119909

119899]

=

infin

sum1198981119898119899=0

(119886)1198981+sdotsdotsdot+119898

119899

(1198871)1198981

sdot sdot sdot (119887119899)119898119899

(119888)1198981+sdotsdotsdot+119898

119899

1199091198981

1


119909119898119899

119899

119898119899

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091198991003816100381610038161003816 lt 1)

(62)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119896

(119911119896minus1)120572

sdot119911119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) (119911119896minus1

)120572

sdot

119896minus1

prod119904=1

(1 minus119911

119908eminus2120587119894119904119896)


1003816100381610038161003816100381610038161003816100381610038161003816119908=119911

(63)




119911119896119874120572

120573Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= Φ(12058811205881199011119896120590

11205901199021119896)

12058211205821199011205721205831120583119902120573

(119911 119904 119886 119887 120582)

(64)



119896

(119911119896minus1)120572

119911

119874120572

120573

times

1

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119911119899+(119896minus1)120572

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


sdot((119911119908) eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119908=119911

=119896

120582Γ (119904) (119911119896minus1)120572

times

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

119899(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981



sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

times119911

119874120572

120573119911


119896minus1+119899+(119896minus1)120572

=119896

120582Γ (119904)

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119886 + 119899)119904

prod119902

119895=1(120583

119895)120590119895119899

11986720

02

times [(119886 + 119899) 1198871120582

| (119904 1) (01

120582)]

119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(1 + 120572 minus 120573)1198981


times(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

sdotΓ (120573) Γ (119898


119896minus1+ 119899 + 119896120572)

Γ (120572) Γ (120573 + 1198981+ sdot sdot sdot + 119898

119896minus1+ 119899 + 119896120572)


=119896Γ (120573) Γ (119896120572)

120582Γ (119904) Γ (120572) Γ (120573 + (119896 minus 1) 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)120588119895119899

(119896120572)11989911986720

02[(119886+119899) 119887

1120582 | (119904 1) (0 1120582)]

(119886 + 119899)119904

sdot prod119902

119895=1(120583

119895)120590119895119899

(120573 + (119896 minus 1) 120572)119899

times119911119899

119899

sdot

infin

sum1198981119898119896minus1

=0

(119899+119896120572)1198981+sdotsdotsdot+119898

119896minus1

(1+120572minus120573)1198981



119896minus1

sdot(eminus2120587119894119896)

1198981

1198981


119898119896minus1

119898119896minus1

(65)



11205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


119911minus1198881198990(119911 minus 119911

0)119888119899

Γ (119888119899 + 1) Γ (1 minus 119888119899)

sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(1003816100381610038161003816119911 minus 1199110

1003816100381610038161003816 =100381610038161003816100381611991101003816100381610038161003816 120582 gt 0)

(66)


Proof Setting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902


0 we

have

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888

infin


(119911 minus 1199110)119888119899

Γ (1 + 119888119899)

sdot119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

(67)


0| = |119911

0|


119863119888119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199110

=119911minus1198881198990

Γ (1 minus 119888119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus119888119899

(1199110 119904 119886 119887 120582)

(68)



Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(69)


119911 =1199111

2+1199111



Proof By taking 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 9 with 119911

2= 0 120583 = 120572 ] = 120573 and 0 lt 119888 ≦ 1 we find

that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888(119911 minus 1199111)minus120573

119911minus120572

1199111

sdot

infin



sdot (119911 minus 119911

1

119911)119888119899+120574

sdot119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

(71)



119863minus120573+119888119899+120574

119911119911

120572+119888119899+120574minus1

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120572+120573minus1

1

Γ (120572 + 119888119899 + 120574)

Γ (120572 + 120573)

times Φ(1205881120588119901112059011205901199021)

1205821120582119901120572+119888119899+120574120583

1120583119902120572+120573

(1199111 119904 119886 119887 120582)

(72)




Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573+120574

(119911 minus 1199111)minus120572+120574

119911120573+120572minus2120574minus1

1

sdot

infin


sin [(120573 minus 119888119899 minus 120574) 120587] e119894120587119888(119899+1)

sin [(120573 + 119888 minus 120574) 120587] Γ (1 minus 120572 + 119888119899 + 120574)

sdot (119911(119911 minus 119911

1)

11991121

)

119888119899

Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdot [(119911 minus 1199111)Φ

(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (120573 minus 119888119899 minus 120574

120572 + 120573 minus 2119888119899 minus 2120574) 119911

1

sdotΦ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)]

(73)


119911 =1199111

2+1199111



Proof Putting 119891(119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582) in Theo-rem 10 with 119911

2= 0 0 lt 119888 ≦ 1 119902(119911) = 1 and 120579(119911) =

(119911 minus 1199111)(119911 minus 119911

2) we find that

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

= 119888119911minus120573

(119911 minus 1199111)minus120572

sdot

infin



e119894120587119888(119899+1)[119911(119911 minus 1199111)]119888119899+120574

Γ (1 minus 120572 + 119888119899 + 120574)

sdot 119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

(75)


119863minus120572+119888119899+120574

119911


(119911+119908minus1199111)Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816 119911=1199111(119908=119911)

= 119863minus120572+119888119899+120574

119911119911

120573minus119888119899minus120574

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

+ (119911 minus 1199111)119863

minus120572+119888119899+120574

119911


Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)100381610038161003816100381610038161003816119911=1199111

= 119911120573+120572minus2119888119899minus2120574

1


Γ (1 + 120573 + 120572 minus 2119888119899 minus 2120574)

sdot Φ(1205881120588119901112059011205901199021)

12058211205821199011+120573minus119888119899minus120574120583

11205831199021+120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582)

+ (119911 minus 119911

1

1199111

)Γ (120573 minus 119888119899 minus 120574)

Γ (120573 + 120572 minus 2119888119899 minus 2120574)

sdotΦ(1205881120588119901112059011205901199021)

1205821120582119901120573minus119888119899minus120574120583

1120583119902120573+120572minus2119888119899minus2120574

(1199111 119904 119886 119887 120582) )

(76)



(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

=Γ (120591) Γ (1 + ] minus 120591) sin (120574120587)

120587

sdot

infin


(minus1)119899

Φ(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)


(77)



12058211205821199011205831120583119902


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=

infin


(] minus 120591120574 + 119899


119911]minus1

sdot 119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(78)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)



119911]minus1 =

Γ (])Γ (120591 + 120574 + 119899)

119911120591+120574+119899minus1

119863120574+119899

119911Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=119911minus120574minus119899

Γ (1 minus 120574 minus 119899)Φ

(1205881120588119901112059011205901199021)

1205821120582119901112058311205831199021minus120574minus119899

(119911 119904 119886 119887 120582)

(79)



Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

= (Γ (120591) Γ (1 + ] minus 120591) sin (120573120587)




sdotsin (120579120587)

sin [(120579 + 120574) 120587]

infin



sdot Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(80)



120572 = ] minus 120591 119906 (119911) = 119911120591minus120573

V (119911) = Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(81)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)


sdot

infin


119863]minus120591+120574+1minus119899119911




119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

(82)


119863]minus120591119911

119911]minus1Φ

(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (])Γ (120591)

119911120591minus1

Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886 119887 120582)

119863]minus120591+120574+1minus119899119911


=Γ (] minus 120579 minus 119899)


119863minus1minus120574+119899

119911119911

120579minus1+119899

Φ(12058811205881199011205901120590119902)

12058211205821199011205831120583119902

(119911 119904 119886 119887 120582)

=Γ (120579 + 119899)

Γ (1 + 120579 + 120574)

times 119911120579+120574

Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120574

(119911 119904 119886 119887 120582)

(83)





lim119887rarr0

11986720

02[(119886 + 119899) 119887

1120582

| (119904 1) (01

120582)] = 120582Γ (119904)

(120582 gt 0)

(84)



Φ(120588112058811990113120590

112059011990213)

12058211205821199011205721205831120583119902120573

(119911 119904 119886)

=3Γ (120573) Γ (3120572)

Γ (120572) Γ (120573 + 120572)

sdot

infin

sum119899=0

prod119901

119895=1(120582

119895)119899120588119895

119899prod119902

119895=1(120583

119895)119899120590119895

(3120572)119899

(120573 + 120572)119899

119911119899

(119899 + 119886)119904


(85)


1198651[119886 119887

1 119887

2 119888 119909

1 119909

2]

=

infin

sum11989811198982=0

(119886)1198981+1198982

(1198871)1198981

(1198872)1198982

(119888)1198981+1198982

1199091198981

1

1198981

1199091198982

2

1198982

(max 100381610038161003816100381611990911003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 lt 1)

(86)




1= 120583 in



Θ120582

120583(119911 119904 119886 119887)

= 119888119911minus120572

(119911 minus 1199111)minus120573

119911120572+120573

1

sdot

infin



sdot Φ(111)

120583120572+119888119899+120574120572+120573(119911

1 119904 119886 119887 120582) (

119911 minus 1199111

119911)119888119899+120574

(87)


119911 =1199111

2+1199111





Φ(1205881120588119901112059011205901199021)

1205821120582119901]1205831120583119902120591(119911 119904 119886)

=Γ (120591) Γ (1 + ] minus 120591)



times (Γ (1 + 120574 + 120579) sin [(] minus 120591 + 120573) 120587]


sdot

infin


Γ (]minus120579minus119899) Γ (120579+119899)Φ(1205881120588119901112059011205901199021)

1205821120582119901120579+119899120583

11205831199021+120579+120582

(119911 119904 119886)





References












































Documents

Recent Trends in Special Numbers and Special Functions and