Research Article Certain Class of Generating …downloads.hindawi.com › journals › aaa › 2014 › 714560.pdfhypergeometric function (",# N 0)whoseparticular cases are known to

Research ArticleCertain Class of Generating Functions forthe Incomplete Hypergeometric Functions

Junesang Choi1 and Praveen Agarwal2

1 Department of Mathematics Dongguk University Gyeongju 780-714 Republic of Korea2Department of Mathematics Anand International College of Engineering Jaipur 303012 India

Correspondence should be addressed to Junesang Choi junesangmaildonggukackr

Received 4 June 2014 Accepted 17 July 2014 Published 18 August 2014

Academic Editor Ali H Bhrawy

Copyright copy 2014 J Choi and P Agarwal 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

Generating functions play an important role in the investigation of various useful properties of the sequences which they generateIn this paper we aim to establish certain generating functions for the incomplete hypergeometric functions introduced by Srivastavaet al (2012) All the derived results in this paper are general and can yield a number of (known and new) results in the theory ofgenerating functions

1 Introduction and Definitions

A lot of research work has recently come up on the studyand development of the familiar incomplete Gamma typefunctions like 120574(119904 119909) and Γ(119904 119909) given in ((1)) and ((2))respectively The study of incomplete Gamma functions hasa very long history (see eg [1]) and now stands on fairlyfirm footing through the research contributions of variousauthors (see eg [2ndash17]) Incomplete Gamma functions areimportant special functions and their closely related onesare widely used in physics and engineering therefore theyare of interest to physicists engineers statisticians andmathematicians

The theory of the incomplete Gamma functions as apart of the theory of confluent hypergeometric functions hasreceived its first systematic exposition by Tricomi [18] in theearly 1950sThe familiar incomplete Gamma functions 120574(119904 119909)and Γ(119904 119909) are defined respectively by

120574 (119904 119909) = int

119909

0

119905119904minus1

119890minus119905

119889119905 (R (119904) gt 0 119909 ge 0) (1)

Γ (119904 119909) = int

infin

119909

119905119904minus1

119890minus119905

119889119905 (119909 ge 0 R (119904) gt 0 when 119909 = 0)

(2)

The following decomposition formula holds

120574 (119904 119909) + Γ (119904 119909) = Γ (119904) (R (119904) gt 0) (3)

where Γ(119904) is the familiar Gamma function defined by

Γ (119904) = int

infin

0

119905119904minus1

119890minus119905

119889119905 (R (119904) gt 0) (4)

Historically ((1)) and ((2)) were first studied in 1877 for119909 = 1 by Prym [1] The functions ((1)) and ((2)) are alsoreferred to as Prymrsquos functions For general 119909 gt 0 (even for119909 lt 0) the function ((2)) appears in Exercises de CalculIntegral by Legendre [19] and in some of his later works

The function Γ(119904 119909) can be expressed in terms of Tricomirsquosconfluent hypergeometric function Ψ(119886 119888 119909) as follows (see[6 page 266 Equation 692(21)])

Γ (119904 119909) = 119890minus119909

Ψ (1 minus 119904 1 minus 119904 119909) (5)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 714560 5 pageshttpdxdoiorg1011552014714560

2 Abstract and Applied Analysis

In terms of the Gamma function Γ(119911) Pochhammersymbol (120582)

119899is defined (for 120582 isin C) by (see eg [10 page 2

and pages 4ndash6])

(120582)119899=

1 (119899 = 0)

120582 (120582 + 1) sdot sdot sdot (120582 + 119899 minus 1) (119899 isin N = 1 2 3 )

=Γ (120582 + 119899)

Γ (120582) (120582 isin C Z

minus

0)

(6)

where C and Zminus0denote the sets of complex numbers and

nonpositive integers respectivelyRecently Srivastava et al [16] introduced and studied

some fundamental properties and characteristics of a familyof the following twopotentially useful generalized incompletehypergeometric functions defined as follows

119901120574119902[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=

infin

sum

119899=0

(1198861 119909)119899(1198862)119899sdot sdot sdot (119886119901)119899

(1198871)119899sdot sdot sdot (119887119902)119899

sdot119911119899

119899

(7)

119901Γ119902[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=

infin

sum

119899=0

[1198861 119909]119899(1198862)119899sdot sdot sdot (119886119901)119899

(1198871)119899sdot sdot sdot (119887119902)119899

sdot119911119899

119899

(8)

where (1198861 119909)119899and [119886

1 119909]119899are certain interesting generaliza-

tions of the Pochhammer symbol (120582)119899which are defined in

terms of the incomplete Gamma type functions 120574(119904 119909) andΓ(119904 119909) given in ((1)) and ((2)) by

(120582 119909)] =120574 (120582 + ] 119909)

Γ (120582) (120582 ] isin C 119909 ge 0)

[120582 119909]] =Γ (120582 + ] 119909)

Γ (120582) (120582 ] isin C 119909 ge 0)

(9)

These incomplete Pochhammer symbols (120582 119909)] and[120582 119909]] which were defined by Srivastava et al [16] like ((3))also satisfy the following decomposition relation

(120582 119909)] + [120582 119909]] = (120582)] (120582 ] isin C 119909 ge 0) (10)

Remark 1 As already mentioned by Srivastava et al [16Remark 7] (see also [17 page 3220 Remark]) since

1003816100381610038161003816(120582 119909)1198991003816100381610038161003816 le

1003816100381610038161003816(120582)1198991003816100381610038161003816

1003816100381610038161003816[120582 119909]1198991003816100381610038161003816 le

1003816100381610038161003816(120582)1198991003816100381610038161003816

(119899 isin N 120582 isin C 119909 ge 0)

(11)

the precise (sufficient) conditions under which the infiniteseries in the definitions ((7)) and ((8)) would converge abso-lutely can be derived from those that are well-documented

in the case of the generalized hypergeometric function119901119865119902

(119901 119902 isin N) (see for details [20 pages 72-73] and [11page 20] see also [21ndash23]) Indeed in their special casewhen 119909 = 0 both

119901120574119902(119901 119902 isin N) and

119901Γ119902(119901 119902 isin N)

would reduce immediately to the extensively investigatedgeneralized hypergeometric function

119901119865119902(119901 119902 isin N) (see eg

[20 Chapter 5] see also [10 Section 15]) Furthermore as animmediate consequence of the definitions ((7)) and ((8)) wehave the following decomposition formula

119901120574119902[(1198861 119909) 1198862 119886

119901

1198871 119887

119902

119911]

+119901Γ119902[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902

119911]

=119901119865119902[1198861 119886

119901

1198871 119887

119902119911]

(12)

in terms of the familiar generalized hypergeometric function119901119865119902(119901 119902 isin N)

Generating functions play an important role in the inves-tigation of various useful properties of the sequences whichthey generate They are used in finding certain propertiesand formulas for numbers and polynomials in a wide varietyof research subjects indeed in modern combinatorics Fora systematic introduction to and several interesting (anduseful) applications of the various methods of obtaininglinear bilinear bilateral or mixed multilateral generatingfunctions for a fairly wide variety of sequences of specialfunctions (and polynomials) in one two and more variablesamong much abundant literature we refer to the extensiveworks by Srivastava andManocha [24] andAgarwal andKoul[25] In this regard in fact a remarkably large number ofgenerating functions involving a variety of special functionshave been developed by many authors (see eg [24 26]see also [27]) Also many generating functions containingthe incomplete hypergeometric functions ((7)) and ((8)) havebeen presented (see eg [17 Corollary 3]) Here motivatedmainly by the works of both Chen and Srivastava [28]and Srivastava and Cho [17] we present certain generatingfunctions involving the incomplete hypergeometric functions((7)) and ((8)) Furthermore it should be mentioned inpassing that our results in the present paper are establishedby using a different method employed by [17]

2 Generating Functions for the IncompleteHypergeometric Functions

In this section we establish certain generating functionsfor the incomplete hypergeometric functions ((7)) and ((8))asserted byTheorem 2

Abstract and Applied Analysis 3

Theorem 2 The following generating functions hold trueinfin

sum

119896=0

(120572 + 119896 minus 1

119896)119901120574119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901120574119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(119909 ge 0 120572 isin C |119905| lt 1)

(13)

infin

sum

119896=0

(120572 + 119896 minus 1

119896)119901Γ119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901Γ119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(119909 ge 0 120572 isin C |119905| lt 1)

(14)

Proof For convenience let the left-hand side of ((13)) bedenoted by S Applying the series expression of ((7)) to Swe get

S =

infin

sum

119896=0

(120572 + 119896 minus 1

119896)

sdot (

infin

sum

119899=0

(1198861 119909)119899(1198862)119899sdot sdot sdot (119886119901)119899

(1 minus 120572 minus 119896)119899(1198871)119899sdot sdot sdot (119887119902)119899

sdot119911119899

119899) 119905119896

(15)

Using the following known identities (see eg [10 page 5])

(120572

119899) =

Γ (120572 + 1)

119899Γ (120572 minus 119899 + 1) (119899 isin N

0 120572 isin C)

Γ (120572 minus 119899)

Γ (120572)=

(minus1)119899

(1 minus 120572)119899

(119899 isin N0 120572 isin C Z)

(16)

Z being the set of integers and N0= N cup 0 we can prove

the following identity (see [28 page 169])

(1 minus 120572 minus 119896)119899= (1 minus 120572)

119899(120572 + 119896 minus 1

119896)

sdot (120572 minus 119899 + 119896 minus 1

119896)

minus1

(119896 119899 isin N0)

(17)

By changing the order of summations in ((15)) and using theidentity ((17)) after little simplification we have

S =

infin

sum

119899=0

(1198861 119909)119899(1198862)119899sdot sdot sdot (119886119901)119899

(1 minus 120572)119899 (1198871)119899sdot sdot sdot (119887119902)119899

sdot119911119899

119899

infin

sum

119896=0

(120572 minus 119899 + 119896 minus 1

119896) 119905119896

(18)

We find that the inner sum in ((18)) is the generalizedbinomial expansioninfin

sum

119896=0

(120572 minus 119899 + 119896 minus 1

119896) 119905119896

= (1 minus 119905)minus120572+119899

(|119905| lt 1) (19)

Finally replacing the inner sum of ((18)) by the identity ((19))yields our desired result ((13))

It is easy to see that a similar argument as in the proof of((13)) will establish the result ((14)) This completes the proofof Theorem 2

Remark 3 Recently Srivastava and Cho [17] presented avery general class of certain interesting generating functionsinvolving the incomplete hypergeometric functions ((7)) and((8)) by essentially using the following interesting and usefulunified expansion formula given by Gould (see [29 page 196Equation (9)] see also [17 page 3221])

infin

sum

119896=0

120581

120581 + (120573 + 1) 119899(120572 + (120573 + 1) 119899

119899) 119905119899

= (1 + 120585)120572

infin

sum

119899=0

(minus1)119899

(120572 minus 120581

119899)(

119899 +120581

120573 + 1

119899minus1

)(120585

1 + 120585)

119899

(20)

where 120572 120573 and 120581 are complex numbers independent of 119899 and120585 is a function of 119905 defined implicitly by

120585 = 119905(1 + 120585)120573+1

120577 (0) = 0

(21)

The results [17 Corollary 3] look very similar to those giveninTheorem 2 Yet it is easy to see that they cannot be specialor general cases of the other onersquos results

3 Further Generalization of theGenerating Functions for the IncompleteHypergeometric Functions

A further generalization of the incomplete hypergeometricfunctions ((7)) and ((8)) is given in the following definition

Definition 4 Let us introduce two sequences Λ(120572120573)119896

(119911)infin

119896=0

and Ω(120572120573)119896

(119911)infin

119896=0defined by

Λ(120572120573)119896

(119911) = Λ(120572120573)119896

[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=119901120574119902+120573

[(1198861 119909) 119886

2 119886

119901

Δ (120573 1 minus 120572 minus 119896) 1198871 119887

119902119911]

(119909 ge 0 120573 isin N 120572 isin C)

Ω(120572120573)119896

(119911) = Ω(120572120573)119896

[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=119901Γ119902+120573

[(1198861 119909) 119886

2 119886

119901

Δ (120573 1 minus 120572 minus 119896) 1198871 119887

119902119911]

119909 (119909 ge 0 120573 isin N 120572 isin C)

(22)


where for convenience Δ(120573 120572) abbreviates the array of 119873parameters as follows

120572

120573120572 + 1

120573

120572 + 120573 minus 1

120573 (120572 isin C 120573 isin N) (23)

Then as inTheorem 2 we can give the following generat-ing functions for the generalized incomplete hypergeometricfunctions ((22)) asserted byTheorem 5

Theorem 5 Each of the following identities holds true

infin

sum

119896=0

(120572 + 119898 + 119896 minus 1

119896)Λ

(120572120573)119898+119896

(119911) 119905119896

= (1 minus 119905)minus120572minus119898

Λ(120572120573)119898

(119911(1 minus 119905)120573

)

(119909 ge 0 119898 isin N0 120572 isin C 120573 isin N |119905| lt 1)

(24)

infin

sum

119896=0

(120572 + 119898 + 119896 minus 1

119896)Ω

(120572120573)119898+119896

(119911) 119905119896


Ω(120572120573)119898

(119911(1 minus 119905)120573

)

(25)

(119909 ge 0 119898 isin N0 120572 isin C 120573 isin N |119905| lt 1) (26)

Proof Similarly as in Theorem 2 we can prove the results inTheorem 5 So their details are left to the interested readerby instead of the essential identity ((17)) presenting thefollowing identity

(1 minus 120572 minus 119898 minus 119896)120573119899= (1 minus 120572 minus 119898)

120573119899(120572 + 119898 + 119896 minus 1

119896)

sdot (120572 + 119898 minus 120573119899 + 119896 minus 1

119896)

minus1

(119896 119899 isin N0 120573 isin N)

(27)

It should be noted that if we set 120573 = 1 and replace 120572 by120572 minus 119898 in ((24)) we are easily led to the result ((13))

Concluding Remarks If we add the two generating functions((13)) and ((14)) and use the decomposition formula ((11)) wehave an interesting result expressed in terms of generalizedhypergeometric functions

119901119865119902

infin

sum

119896=0

(120572 + 119896 minus 1

119896)119901119865119902+1

[1198861 1198862 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901119865119902+1

[1198861 1198862 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(120572 isin C |119905| lt 1)

(28)

We also observe that the result ((28)) corresponds to thatgiven in [28 page 170 Equation (512)]

The generalized incomplete hypergeometric functionsgiven in ((7)) and ((8)) reduce when 119909 = 0 to the generalizedhypergeometric function

119901119865119902(119901 119902 isin N

0) whose particular

cases are known to express most of the special functionsoccurring in the mathematical physical and engineeringsciences Therefore most of the known and widely investi-gated special functions are expressible also in terms of thegeneralized incomplete hypergeometric functions

119901120574119902(119901 119902 isin

N0) and

119901Γ119902(119901 119902 isin N

0) (for some interesting examples

and applications see [16 Sections 5 and 6]) In view ofthis observation the results presented here being of generalcharacter can yield numerous generating functions for acertain class of incomplete hypergeometric polynomials (see[17]) and other special functions which are expressible interms of hypergeometric functions Finally we conclude ourpresent investigation by remarking that our results presentedhere are also believed to give some contribution to thecommunication theory probability theory and groundwaterpumping modeling

Conflict of Interests

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

Acknowledgments

The authors should express their deep gratitude to all thereferees for their very helpful and critical comments origi-nating from only detailed reviews of this paper by sharingtheir valuable time This research was in part supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea funded by the Ministry ofEducation Science and Technology of the Republic of Korea(Grant no 2010-0011005)This work was supported by Dong-guk University Research Fund

References

[1] F E Prym ldquoZur theorie der gamma functionrdquo The Journal furdie Reine und Angewandte Mathematik vol 82 pp 165ndash1721877

[2] M Abramowitz and I A Stegun EdsHandbook ofMathemati-cal Functions with Formulas Graphs and Mathematical TablesApplied Mathematics Series 55 National Bureau of StandardsWashington DC USA 9th edition 1984

[3] P Agarwal ldquoCertain properties of the generalized Gauss hyper-geometric functionsrdquo Applied Mathematics amp Information Sci-ences vol 8 no 5 pp 2315ndash2320 2014

[4] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and its Applications CambridgeUniversity Press Cambridge UK 1999

[5] E H Doha and W M Abh-Elhameed ldquoNew linearization for-mulae for the products of Chebyshev polynomials of third andfourth kindrdquo Rocky Mountain Journal of Mathematics In press

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 1 McGraw-Hill BookCompany New York NY USA 1953


[7] N L Johnson S Kotz and N Balakrishnan Continuous Uni-variate Distributions vol 2 John Wiley amp Sons New York NYUSA 1995

[8] W Koef Hypergepometric Summation Vieweg Braunschweig-Wiebaden 1998

[9] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic Publishers DordrechtThe Netherlands 2001

[10] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[11] H M Srivastava and P W Karlsson Multiple Gaussian Hyper-geometric Series Ellis Horwood Chichester UK 1985

[12] H M Srivastava and B R K Kashyap Special Functions inQueuing Theory and Related Stochastic Processes AcademicPress New York NY USA 1982

[13] N M Temme Special Functions An Introduction to ClassicalFunctions of Mathematical Physics John Wiley and Sons NewYork NY USA Brisbane and Toronto Chichester UK 1996

[14] G NWatsonATreatise on theTheory of Bessel Functions Cam-bridge University Press Cambridge UK 2nd edition 1944

[15] E TWhittaker andG NWatsonACourse ofModern AnalysisAn Introduction to the GeneralTheory of Infinite Processes and ofAnalytic Functions With an Account of the Principal Transcen-dental Functions Cambridge University Press New York NYUSA 4th edition 1963

[16] H M Srivastava M A Chaudhry and R P Agarwal ldquoTheincomplete Pochhammer symbols and their applications tohypergeometric and related functionsrdquo Integral Transforms andSpecial Functions vol 23 no 9 pp 659ndash683 2012

[17] R Srivastava and N E Cho ldquoGenerating functions for a cer-tain class of incomplete hypergeometric polynomialsrdquo AppliedMathematics and Computation vol 219 no 6 pp 3219ndash32252012

[18] F G Tricomi ldquoSulla funzione gamma incompletardquo Annali diMatematica Pura ed Applicata vol 31 no 1 pp 263ndash279 1950

[19] A M Legendre Exercises de Calcul Integral Sur Divers Ordersde Transcendantes et Sur les Quadratures vol 1 Courcier ParisFrance 1981

[20] E D Rainville Special Functions Macmillan New York NYUSA 1960

[21] B C Carlson Special Functions of Applied Mathematics Aca-demic Press New York NY USA 1977

[22] Y L Luke Mathematical Functions and Their ApproximationsAcademic Press New York NY USA 1975

[23] L J Slater Generalized Hypergeometric Functions CambridgeUniversity Press Cambridge UK 1966

[24] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Ellis Horwood John Wiley amp SonsNew York NY USA 1984

[25] P Agarwal and C L Koul ldquoOn generating functionsrdquo Journalof Rajasthan Academy of Physical Sciences vol 2 no 3 pp 173ndash180 2003

[26] D Zeitlin ldquoA new class of generating functions for hypergeo-metric polynomialsrdquo Proceedings of the AmericanMathematicalSociety vol 25 pp 405ndash412 1970

[27] H M Srivastava ldquoSome bilateral generating functions for aclass of special functions I and IIrdquo Indagationes Mathematicae(Proceedings) vol 83 no 2 pp 234ndash246 1980

[28] M-P Chen and H M Srivastava ldquoOrthogonality relations andgenerating functions for Jacobi polynomials and related hyper-geometric functionsrdquo Applied Mathematics and Computationvol 68 no 2-3 pp 153ndash188 1995

[29] H W Gould ldquoA series transformation for finding convolutionidentitiesrdquo Duke Mathematical Journal vol 28 pp 193ndash2021961

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In terms of the Gamma function Γ(119911) Pochhammersymbol (120582)

119899is defined (for 120582 isin C) by (see eg [10 page 2

and pages 4ndash6])

(120582)119899=

1 (119899 = 0)

120582 (120582 + 1) sdot sdot sdot (120582 + 119899 minus 1) (119899 isin N = 1 2 3 )

=Γ (120582 + 119899)

Γ (120582) (120582 isin C Z

minus

0)

(6)

where C and Zminus0denote the sets of complex numbers and

nonpositive integers respectivelyRecently Srivastava et al [16] introduced and studied

some fundamental properties and characteristics of a familyof the following twopotentially useful generalized incompletehypergeometric functions defined as follows

119901120574119902[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=

infin

sum

119899=0

(1198861 119909)119899(1198862)119899sdot sdot sdot (119886119901)119899

(1198871)119899sdot sdot sdot (119887119902)119899

sdot119911119899

119899

(7)

119901Γ119902[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=

infin

sum

119899=0

[1198861 119909]119899(1198862)119899sdot sdot sdot (119886119901)119899

(1198871)119899sdot sdot sdot (119887119902)119899

sdot119911119899

119899

(8)

where (1198861 119909)119899and [119886

1 119909]119899are certain interesting generaliza-

tions of the Pochhammer symbol (120582)119899which are defined in

terms of the incomplete Gamma type functions 120574(119904 119909) andΓ(119904 119909) given in ((1)) and ((2)) by

(120582 119909)] =120574 (120582 + ] 119909)

Γ (120582) (120582 ] isin C 119909 ge 0)

[120582 119909]] =Γ (120582 + ] 119909)

Γ (120582) (120582 ] isin C 119909 ge 0)

(9)

These incomplete Pochhammer symbols (120582 119909)] and[120582 119909]] which were defined by Srivastava et al [16] like ((3))also satisfy the following decomposition relation

(120582 119909)] + [120582 119909]] = (120582)] (120582 ] isin C 119909 ge 0) (10)

Remark 1 As already mentioned by Srivastava et al [16Remark 7] (see also [17 page 3220 Remark]) since

1003816100381610038161003816(120582 119909)1198991003816100381610038161003816 le

1003816100381610038161003816(120582)1198991003816100381610038161003816

1003816100381610038161003816[120582 119909]1198991003816100381610038161003816 le

1003816100381610038161003816(120582)1198991003816100381610038161003816

(119899 isin N 120582 isin C 119909 ge 0)

(11)

the precise (sufficient) conditions under which the infiniteseries in the definitions ((7)) and ((8)) would converge abso-lutely can be derived from those that are well-documented

in the case of the generalized hypergeometric function119901119865119902

(119901 119902 isin N) (see for details [20 pages 72-73] and [11page 20] see also [21ndash23]) Indeed in their special casewhen 119909 = 0 both

119901120574119902(119901 119902 isin N) and

119901Γ119902(119901 119902 isin N)

would reduce immediately to the extensively investigatedgeneralized hypergeometric function

119901119865119902(119901 119902 isin N) (see eg

[20 Chapter 5] see also [10 Section 15]) Furthermore as animmediate consequence of the definitions ((7)) and ((8)) wehave the following decomposition formula

119901120574119902[(1198861 119909) 1198862 119886

119901

1198871 119887

119902

119911]

+119901Γ119902[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902

119911]

=119901119865119902[1198861 119886

119901

1198871 119887

119902119911]

(12)

in terms of the familiar generalized hypergeometric function119901119865119902(119901 119902 isin N)

Generating functions play an important role in the inves-tigation of various useful properties of the sequences whichthey generate They are used in finding certain propertiesand formulas for numbers and polynomials in a wide varietyof research subjects indeed in modern combinatorics Fora systematic introduction to and several interesting (anduseful) applications of the various methods of obtaininglinear bilinear bilateral or mixed multilateral generatingfunctions for a fairly wide variety of sequences of specialfunctions (and polynomials) in one two and more variablesamong much abundant literature we refer to the extensiveworks by Srivastava andManocha [24] andAgarwal andKoul[25] In this regard in fact a remarkably large number ofgenerating functions involving a variety of special functionshave been developed by many authors (see eg [24 26]see also [27]) Also many generating functions containingthe incomplete hypergeometric functions ((7)) and ((8)) havebeen presented (see eg [17 Corollary 3]) Here motivatedmainly by the works of both Chen and Srivastava [28]and Srivastava and Cho [17] we present certain generatingfunctions involving the incomplete hypergeometric functions((7)) and ((8)) Furthermore it should be mentioned inpassing that our results in the present paper are establishedby using a different method employed by [17]

2 Generating Functions for the IncompleteHypergeometric Functions

In this section we establish certain generating functionsfor the incomplete hypergeometric functions ((7)) and ((8))asserted byTheorem 2



sum

119896=0

(120572 + 119896 minus 1

119896)119901120574119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901120574119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(119909 ge 0 120572 isin C |119905| lt 1)

(13)

infin

sum

119896=0

(120572 + 119896 minus 1

119896)119901Γ119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901Γ119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(119909 ge 0 120572 isin C |119905| lt 1)

(14)


S =

infin

sum

119896=0

(120572 + 119896 minus 1

119896)

sdot (

infin

sum

119899=0

(1198861 119909)119899(1198862)119899sdot sdot sdot (119886119901)119899


sdot119911119899

119899) 119905119896

(15)


(120572

119899) =

Γ (120572 + 1)

119899Γ (120572 minus 119899 + 1) (119899 isin N

0 120572 isin C)

Γ (120572 minus 119899)

Γ (120572)=

(minus1)119899

(1 minus 120572)119899

(119899 isin N0 120572 isin C Z)

(16)




119899(120572 + 119896 minus 1

119896)

sdot (120572 minus 119899 + 119896 minus 1

119896)

minus1

(119896 119899 isin N0)

(17)


S =

infin

sum

119899=0

(1198861 119909)119899(1198862)119899sdot sdot sdot (119886119901)119899


sdot119911119899

119899

infin

sum

119896=0

(120572 minus 119899 + 119896 minus 1

119896) 119905119896

(18)


sum

119896=0

(120572 minus 119899 + 119896 minus 1

119896) 119905119896

= (1 minus 119905)minus120572+119899

(|119905| lt 1) (19)




infin

sum

119896=0

120581

120581 + (120573 + 1) 119899(120572 + (120573 + 1) 119899

119899) 119905119899

= (1 + 120585)120572

infin

sum

119899=0

(minus1)119899

(120572 minus 120581

119899)(

119899 +120581

120573 + 1

119899minus1

)(120585

1 + 120585)

119899

(20)


120585 = 119905(1 + 120585)120573+1

120577 (0) = 0

(21)





(119911)infin

119896=0

and Ω(120572120573)119896

(119911)infin

119896=0defined by

Λ(120572120573)119896

(119911) = Λ(120572120573)119896

[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=119901120574119902+120573

[(1198861 119909) 119886

2 119886

119901

Δ (120573 1 minus 120572 minus 119896) 1198871 119887

119902119911]

(119909 ge 0 120573 isin N 120572 isin C)

Ω(120572120573)119896

(119911) = Ω(120572120573)119896

[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=119901Γ119902+120573

[(1198861 119909) 119886

2 119886

119901

Δ (120573 1 minus 120572 minus 119896) 1198871 119887

119902119911]

119909 (119909 ge 0 120573 isin N 120572 isin C)

(22)



120572

120573120572 + 1

120573

120572 + 120573 minus 1

120573 (120572 isin C 120573 isin N) (23)



infin

sum

119896=0

(120572 + 119898 + 119896 minus 1

119896)Λ

(120572120573)119898+119896

(119911) 119905119896


Λ(120572120573)119898

(119911(1 minus 119905)120573

)


(24)

infin

sum

119896=0

(120572 + 119898 + 119896 minus 1

119896)Ω

(120572120573)119898+119896

(119911) 119905119896


Ω(120572120573)119898

(119911(1 minus 119905)120573

)

(25)




120573119899(120572 + 119898 + 119896 minus 1

119896)

sdot (120572 + 119898 minus 120573119899 + 119896 minus 1

119896)

minus1

(119896 119899 isin N0 120573 isin N)

(27)



119901119865119902

infin

sum

119896=0

(120572 + 119896 minus 1

119896)119901119865119902+1

[1198861 1198862 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901119865119902+1

[1198861 1198862 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(120572 isin C |119905| lt 1)

(28)



119901119865119902(119901 119902 isin N

0) whose particular


119901120574119902(119901 119902 isin

N0) and

119901Γ119902(119901 119902 isin N





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

119896=0

(120572 + 119896 minus 1

119896)119901120574119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901120574119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(119909 ge 0 120572 isin C |119905| lt 1)

(13)

infin

sum

119896=0

(120572 + 119896 minus 1

119896)119901Γ119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901Γ119902+1

[(1198861 119909) 119886

2 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(119909 ge 0 120572 isin C |119905| lt 1)

(14)


S =

infin

sum

119896=0

(120572 + 119896 minus 1

119896)

sdot (

infin

sum

119899=0

(1198861 119909)119899(1198862)119899sdot sdot sdot (119886119901)119899


sdot119911119899

119899) 119905119896

(15)


(120572

119899) =

Γ (120572 + 1)

119899Γ (120572 minus 119899 + 1) (119899 isin N

0 120572 isin C)

Γ (120572 minus 119899)

Γ (120572)=

(minus1)119899

(1 minus 120572)119899

(119899 isin N0 120572 isin C Z)

(16)




119899(120572 + 119896 minus 1

119896)

sdot (120572 minus 119899 + 119896 minus 1

119896)

minus1

(119896 119899 isin N0)

(17)


S =

infin

sum

119899=0

(1198861 119909)119899(1198862)119899sdot sdot sdot (119886119901)119899


sdot119911119899

119899

infin

sum

119896=0

(120572 minus 119899 + 119896 minus 1

119896) 119905119896

(18)


sum

119896=0

(120572 minus 119899 + 119896 minus 1

119896) 119905119896

= (1 minus 119905)minus120572+119899

(|119905| lt 1) (19)




infin

sum

119896=0

120581

120581 + (120573 + 1) 119899(120572 + (120573 + 1) 119899

119899) 119905119899

= (1 + 120585)120572

infin

sum

119899=0

(minus1)119899

(120572 minus 120581

119899)(

119899 +120581

120573 + 1

119899minus1

)(120585

1 + 120585)

119899

(20)


120585 = 119905(1 + 120585)120573+1

120577 (0) = 0

(21)





(119911)infin

119896=0

and Ω(120572120573)119896

(119911)infin

119896=0defined by

Λ(120572120573)119896

(119911) = Λ(120572120573)119896

[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=119901120574119902+120573

[(1198861 119909) 119886

2 119886

119901

Δ (120573 1 minus 120572 minus 119896) 1198871 119887

119902119911]

(119909 ge 0 120573 isin N 120572 isin C)

Ω(120572120573)119896

(119911) = Ω(120572120573)119896

[(1198861 119909) 119886

2 119886

119901

1198871 119887

119902119911]

=119901Γ119902+120573

[(1198861 119909) 119886

2 119886

119901

Δ (120573 1 minus 120572 minus 119896) 1198871 119887

119902119911]

119909 (119909 ge 0 120573 isin N 120572 isin C)

(22)



120572

120573120572 + 1

120573

120572 + 120573 minus 1

120573 (120572 isin C 120573 isin N) (23)



infin

sum

119896=0

(120572 + 119898 + 119896 minus 1

119896)Λ

(120572120573)119898+119896

(119911) 119905119896


Λ(120572120573)119898

(119911(1 minus 119905)120573

)


(24)

infin

sum

119896=0

(120572 + 119898 + 119896 minus 1

119896)Ω

(120572120573)119898+119896

(119911) 119905119896


Ω(120572120573)119898

(119911(1 minus 119905)120573

)

(25)




120573119899(120572 + 119898 + 119896 minus 1

119896)

sdot (120572 + 119898 minus 120573119899 + 119896 minus 1

119896)

minus1

(119896 119899 isin N0 120573 isin N)

(27)



119901119865119902

infin

sum

119896=0

(120572 + 119896 minus 1

119896)119901119865119902+1

[1198861 1198862 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901119865119902+1

[1198861 1198862 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(120572 isin C |119905| lt 1)

(28)



119901119865119902(119901 119902 isin N

0) whose particular


119901120574119902(119901 119902 isin

N0) and

119901Γ119902(119901 119902 isin N





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572

120573120572 + 1

120573

120572 + 120573 minus 1

120573 (120572 isin C 120573 isin N) (23)



infin

sum

119896=0

(120572 + 119898 + 119896 minus 1

119896)Λ

(120572120573)119898+119896

(119911) 119905119896


Λ(120572120573)119898

(119911(1 minus 119905)120573

)


(24)

infin

sum

119896=0

(120572 + 119898 + 119896 minus 1

119896)Ω

(120572120573)119898+119896

(119911) 119905119896


Ω(120572120573)119898

(119911(1 minus 119905)120573

)

(25)




120573119899(120572 + 119898 + 119896 minus 1

119896)

sdot (120572 + 119898 minus 120573119899 + 119896 minus 1

119896)

minus1

(119896 119899 isin N0 120573 isin N)

(27)



119901119865119902

infin

sum

119896=0

(120572 + 119896 minus 1

119896)119901119865119902+1

[1198861 1198862 119886

119901

1 minus 120572 minus 119896 1198871 119887

119902119911] 119905119896

= (1 minus 119905)minus120572

119901119865119902+1

[1198861 1198862 119886

119901

1 minus 120572 1198871 119887

119902119911 (1 minus 119905)]

(120572 isin C |119905| lt 1)

(28)



119901119865119902(119901 119902 isin N

0) whose particular


119901120574119902(119901 119902 isin

N0) and

119901Γ119902(119901 119902 isin N





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Certain Class of Generating …downloads.hindawi.com › journals › aaa › 2014 › 714560.pdfhypergeometric function (",# N 0)whoseparticular cases are known to