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A CERTAIN CLASS OF MULTIPLE GENERATING FUNCTIONS INVOLVING SOME GENERALIZED
SPECIAL FUNCTIONS AND POLYNOMIALS
THESIS SUBMITTED FOR THE AWARD OF THE DEGREE OF
Bottor of fHjilosopijr IN
APPLIED MATHEMATICS
%
BY
NASEEM AHMAD KHAN *'*•'
UNDER THE SUPERVISION OF
PROFESSOR MUMTAZ AHMAD KHAN M Sc., Ph.D. (Lucknow)
DEPARTMENT OF APPLIED MATHEMATICS Z. H. COLLEGE OF ENGINEERING & TECHNOLOGY
FACULTY OF ENGINEERING ALIGARH MUSLIM UNIVERSITY
ALIGARH - 202002 U. P., INDIA
2007
•p, **** t/fc/**--
r ^ ' ; i "
(1 o 3
/
T6770
5 ~T~r r r
ir"
/ fe#m m tfo rc^m^ ofjilmighty Qod, the Most
Cjracious and the Most "Merciful without whose wiCC
this Thesis would not have seen the daylight.
Jill the praise 6e to the almighty Cjod, sustainer
of the worlds, and may the blessings and peace he
upon the Master of (Prophets, on his progeny and alt
his companions.
Hie primary aim of education is to prepare and
qualify the new generation for participation in the
cuCturat development of the people, so that the
succeeding generations may benefit from the
knowledge and experience of their predecessors.
U - l . . 1 ~ Jl_L I 1 L T ^__L : J J I- 1 1 I_JL~TT~-I~Zl l ^ J
2&$
Special Thanks
To
Esteemed Grandparents
MR. N AWA& SHER KHAN
&
MRS. JAMEELA BEGUM
(Dedicated
To My
Esteemed Parents
MOflD. HAMID RAZA KHAN
&
MRS. SAFDRI BEGUM
To
My Beloved Sister
MISS. SALMA RAZA
&
My Dear Brother
DR.WASEEM AHMAD KHAN
Attributed
To My
Beloved Cousin
MISS. SHEEBA
On Her Birth Anniversary
Dated 29 June
With
My Heartfelt
Gratitude and Indebtedness
To
DR. YASHWAM SINGH
d
DR. M. KAMARUJJAMA
PROF. MUMTAZ AHMAD KHAN Chairman
M.ScPh.D.(LUCKNOW) DEPARTMENT OF APPLIED MATHEMATICS ZAKIR HUSAIN COLLEGE OF ENGINEERING & TECHNOLOGY FACULTY OF ENGINEERING, ALIGARH MUSLIM UNIVERSITY ALIGARH - 202002, U.P. INDIA
-ate..2f. .•£.,. .2^2?
CERTIFICATE
Certified that the thesis entitled "Jl Certain Class Of Multiple
generating Functions Involving Some generalizedSpecial Functions
and (Polynomials" is based on a part of research wori^ Mr.Naseem
Ahmad %han carried out under my guidance and supervision in the
(Department of Applied Mathematics, Faculty of Engineering, Jiligarh
Muslim Vniversity, Jlligarh. To the 6est of my knowledge and belief the
worf^includedin the thesis is original and has not been submitted to any
other university or institution for the award of a degree.
This is to further certify that 'Mr. Naseem Jihmad %han has
fulfilled the prescribed conditions of duration and nature given in the
statutes and ordinances ofjlligarh Muslim Vniversity, flligarh.
Vf 4- 20ffl Ctiairm«ri
Applwd Mai£»C**Xt. i > m of E^of .'& Tecto,
4. J.. ALI9ABH
(PROF. MUMTAZ AHMAD KHAN) SUPERVISOR
Ph.No. 0571270027 Mob.No.09897686959
Applied Mathematics A. M. U. Aligarh
CONTENTS Acknowledgment i-v
Preface vi-ix
CHAPTER 1 : PRELIMINARIES
(Introduction, Definitions and Notations) 1-63
1.1 : The H-function 1
1.2 : Definition of H-function 2
1.3: Some Identities of The H-function 5
1.4 : H-function expressed in term of Elementary Special Function 11
1.5 : Elementary Special Function in term of H-function expressed 12
1.6 : The ^-function 13
1.7 : The Multivariable H-function 15
1.8 : Orthogonal Polynomials 20
1.9 : The Classical orthogonal Polynomials 20
1.10: Historical Developments of Fractional calculus 25
1.11: Operators of Fractional Integration 28
1.12 : Fractional Derivatives and Hypergeometric Function 31
1.13 : AT-Fractional DifFerintegral 37
1.14 : The General class of Polynomial Sets 37
1.15 : Fourier Series 40
Applied Mathematics A. M. U. Aligarh
1.16 : The Gamma Function and Related Functions
1.17 : Gaussian Hypergeometric Function and Generalization
1.18 : Hypergeometric function of Two and Several Variables
1.19 : Bessel Function and Hyper Bessel Function
1.20 : Mittag-Leffer's Function and Related Functions
1.21 : Generating Functions
1.21 : Integral Transforms
CHAPTER 2
2.1 :
2.2 :
2.3:
2.4 :
ON SOME FINITE INTEGRALS INVOLVING GENERALIZED POLYNOMIAL SETS AND THE MULTIVARIABLE if-FUNCTION WITH APPLICATIONS 64-90
Introduction
Main Result
CHAPTER 3 :
3.1 :
3.2:
3.3 :
AT-FRACTION,
/ /-FUNCTION
Introduction
Main Result
Special Cases
91-99
41
44
46
51
55
58
61
65
69
Special Cases 81
Applications 85
91
94
96
2
Applied Mathematics A. M. U. Aligarh
CHAPTER 4 : FOURIER SERIES INVOLVING
THE H-FUNCTION 100-124
4.1 : Introduction 100
4.2 : The Integral 106
4.3 : Exponential Fourier Series 108
4.4: Cosine Fourier Series 111
4.5: Sine Fourier Series 114
4.6: Multiple Exponential Fourier Series 116
4.7 : Particular Cases 120
C H A P T E R 5 : INTEGRALS AND FOURIER SERIES INVOLVING
JI-FUNCTION 1 2 5 - 1 4 4
5.1 : Introduction 125
5.2 : The integral 129
5.3 : The Double Integral 133
5.4 : Fourier Series 136
5.5 : Double Fourier Exponential Series 141
% & S*»
Applied Mathematics A. M. U. Aligarh CHAPTER 6 : A FINITE INTEGRAL INVOLVING GENERALIZED
LAURICELLA'S FUNCTION AND THE H-FUNCTION WITH APPLICATION 157-156
6.1 : Introduction 14
6.2 : Main Result 14
6.3 : Particular cases
6.4 : Application
: BIBLIOGRAPHY 157-180
4
Appraises and thanks to the Almighty god who taught the man "what he knew not and
who Blessed him with wisdom and courage. It is His widthat this thesis has seen the daylight.
I would never have words to express my gratitude and indebtedness to my supervisor
(Prof. Mumtaz Ahmad %jian, Chairman, (Department of Applied Mathematics, Afyarh
Muslim University, JiCigarh, under whose continuous guidance, valuable suggestions,
encouragement, inspiration and motivations as wed as his keen interest in my work,, I am able
to complete this Thesis.
I am also highly grateful to (Dr. 'Yashwant Singh and to Or. Mohd. l&marrujjama who
developed in me a sense of analytical approach through their fruitful discussions and
contributions towards the completion of this thesis.
My thanks are also due to (Dr. firman %fian, my M.&hil supervisor, and <Prof. Mohd
Muk$XarA&, T^Chairmen, for their encouragement and good wishes.
I also take this opportunity to thinks other teachers of the department who have
contributed in anyway towards the completion of the worf^
I am also thankful to my seniors (Dr. Abdul 9fakpn %futn presently Senior-Lecturer in
the (Department, (Dr. Mrs. Muqaddds Najmi, (Dr. Afoy %umar Shukla, (Dr. KfiursheedAhmad,
(Dr. ghazi Salama Abukfiammash, (Dr. Abdul (Rahman %fian, (Dr. (Bhagwat Swaroop Sharrna
and (Dr. Mukesh (Pal Singh all of whom obtained their (Ph.(D. degrees under the supervision of
my supervisor for their help and encouragement in completion of my (Ph.(D. work± In
particular, to Mr. (Bijan (Rpuhi another(ph.(D. student of my supervisor who has submitted his
(Ph\(D. thesis recently for his valuable help.
I am thankful to my other seniors Mr. S. M. Amin, Mr. Ajeet TQitnar Sharrna, Mr.
Shak&el Ahmad Alvi, Mr. Kfiursheed Ahmad, Mr. <B. S. Sharrna and Mr. Mohd Asif who
obtained their M.(PhUdegrees under the supervision of my supervisor.
My thanks are also due to Mr. <B. AGdadand Mr. Mohd Asifwho are also pursuing
their (Ph. <D. (Degrees under my supervisor, (Prof. Mumtaz Ahmad%fum.
I would also City to than^my fettow research workers Mr. Mohd Akfjdaq, Mr. Istkirar
Ahmad, Mr. Shakeel Ahmad, Mr. 7C S. JftsarandMr. %fiaGd(Rfifat %fum aH working for
their M.(Phil (Degrees under (Prof Mumtaz Ahmad %fian, who have gone through the
manuscript of this thesis for pointing our errors and omissions in it.
My thanks are also due to other research scholars of the department namely (Dr. (Dinesh
Singh) <Dr. Madhukar Sharma, Mr. (parvez AG, Mr. Aijaz %jrmani, Mr. Naeem Ahmad,
Mr.Saifaraz Mr.Mohd Shahzad, Mr.<Sjfaqat AG, Mr. Waseem Ahmad %jian, Mr. Mohd.
Abbas, Miss. Tarannum %flshmin and Miss. Nagmalrfan for their cooperation.
I also express my thanks to staff members, Mr. Siraj ul TCasan, Mr. Mohd. Tariq,
Mrs.NiswanAG, Mr. ShoaiB Ahmad, Mr. MohdAmin %fian, Mr. Mohd<Parvez, Mr. Mohd
Zahid A& and Mr.Mohd Arf of the (Department of Applied Mathematics, faculty of
'Engineering e£ technology, AGgarh Muslim Vniversity, AGgarh, who have contributed in one
way or other towards the completion of this tas^
Most of the credit of my workjwouldhe to my loving parents, Mr. Hamid^aza %fian and
Mrs. Safdari (Begum whose continuous patience, prayers and blessing on me took, to the
completion of this work* and a significant credit would be to my family members especially my
grand father, Mr. Nawab Sher %fian and grand mother, Mrs.Jameela (Begum, who
encouraged me for pursuing higher qualification.
I ta^e this opportunity to thankjny dear sister Miss. Safma Qfaza, who has completed
M.A- ('English e£ Islamic Studies)from AGgarh Muslim Vniversity, AGgarh and also helped a
lot in my wori^
11
My most affectionate thanks go to my brother Mr. <Waseem Ahmad Kfum, who is
pursuing (BUMS from Jodhpur "University, (Rajasthan, for his encouragement and good
wishes.
My special cordial word of thanks would be to my beloved cousin Miss. Sheeba who
always encouraged, motivated and morally supported me all the way; and enlightened my
spirit in hours of despair with positive thinking.
I shall be failing in my duties if I do not express my gratitude to the wife of my
supervisor Mrs. Aisha (Bono (M.Sc, (B.<Ed), T.g.T., ZakirHusain Model Senior Secondary
School, Migarhfor her kind hospitality and prayer to god for my welfare. I am also thankful
to my supervisor's children Miss Qhazata (Perveen %fian (daughter) M.Tech. ((Electrical), Mr.
(Rflshidlmran Ahmad %fian (son), (B.Tech. H** 'Year (Petro-Chemical), both student of Z. K
College of (Engineering d£ (Technology, JQ..M.V., AGgarh, and to 14 years old Master Abdul
Qiazzaq Kfian (Son), a class IX student of Our Lady Of'Fatima School, ARgarh for enduring
the preoccupation of their father with this work^
I also express my deep appreciation to my beloved elder brother Mr. Syed <Rais Ahmad
'Kashmi and to my sister Mrs. %ahkashan (Parveenfor the happiness shown by them for my
pursuing <Ph.<D. (Degree.
I would also like to thank^my real joint family members, Mr, Abdul gani, Mr.<R&is
Ahmd%fuin, MnShareefAhmad %fuin, Mr. Ahmad <Raza %fian, Mr. SajidAb Xfian, Mr.
<Rashid %fian and their wives Mrs. %fiursheida (Begum, Mrs. Idreesan (Begum, Mrs.
Tabassum, Mrs. !Naushaba <Raza, Mrs. Shakjra (Begum, Mrs. Mahjabeen and their children
Mr. JamalAhmad,, Mr. Ahsan <Rgza, Mr. Salman %juin, Mr. Ifaseen %fian, Miss. Shaheen,
Miss. Shaista, Miss. Wisa, Miss. Mna, Miss, taratuium, Miss. Sabiya, Miss. Mahakjor their
cooperation.
in
My thanks are also due to grand father Mr. (Rflsool 7(fian and their family
members Mr. <Raees Ahmad%fuin, Mr. (Rjyasat Ahmad 7(fian, Mr. (R& Ahmad %fum, Mr.
Sarfaraz Ahmad %jian and their -wives Mrs. Sugra (Begum, Mrs. Qaisar Jahan, Mrs.
Tasfyen (Begum, Mrs Ameer (Bono and their children who prayed for me and supported me
morally.
I am greatly indebted to my guardians Mr. Mohd Irfan his wife Mrs. Najma Irfan
and their famdy members Mrs. Shameem 'Fatima, Mr. Mohd Imran, Mrs. Azra Laeeq, Mr.
Mohd Laeeq, Mrs. <Bushra Azar, Mr. Azhar Islam, Miss. Shee6a Irfan, Mr. Mohd Taizan
and Mr. Mohd Tarhan, Aree6, Shad, Saniya who prayed for me and always supported me
morally.
I also show my gratefulness to Mr. Haji 9fawa6 Hasan, Mrs. Zahida <Parveen and
their famdy members Mrs. Shaista <Parveen, Mr. Mohd Sajid, Mrs. 'Kflhbashan (Parveen, Mr.
ShakgelAhmad, Mr. MasoodHasan, Miss. (Dara&frsan, Mr. Mashfyor Hasan, Mr. Mujahid
Hasan, Miss. Afshan Hawab, Mr. Moha.tsh.im andZoya Hussain who also prayed for me and
supported me morally.
My cordial thanks are also due to Mr. Sayed <Raoof Ahmad, Hashmi Mrs. Sajida
(Begum, and their family members Mr. (Rats AhmadHashmi, Mr. (Bareeq Ahmad Hashmi, Mr.
Zubair Ahmad Hashmi, Mr. Jahnageer, Miss. Shaheen Mr. AurangzeS, Mr. Shaour Ahmad
Hasmi, Mr. Anwar Hashmi, Mr. QulzarHashmiffabarfor their valuable encouragement and
support.
I would also life to thanks to my first teacher (Dr. Mohd TafseerulHaq Xfian and
other teachers Mr. Abdul Wahab, Mr. %afeel Ahmad %fian, Mr.Ak§utTaqi, Mr.Mohd
Anwar, Mr. Afsar Husain Quadri, Mr. Mohsin %fuvn, Mr.togender (Pratap Sigh Chauhan,
Mr.Vtireder Kumar Sharma, (Dr. Vvnod TQimar Qupta, Mr. Jugender Yadav, Abdul <Ra5
%fian, (Dr. (Raza Habib and my uncle Haji Chaudhary 'Hoorul Hasan and his family, Aijaz
iv
Ahmad advocate and their famiCy, <Dr. ShanforLai'andhis family, <Dr. Sharad Chadra Mishra
for their encouragement and good wishes.
I also owe a few words of thanks to my real friends Mr. Jahangeer, Mr. Masood
Hasan, Mr. Mohd Imran, AddSherwani and her wife Mrs. <Farha Steward, Mr. Jfafees
AhmadXfian .Mr.'Wajid AG, Mr-AhmadAG, Mr. MohdTariq, MrASduGdh, Mr. Shahroz
Ahmad, Mr.Shan-e-mansor, Mr.Muddsar, Mr. Satfraz and Mr. Aneesuddin and to my cousin
Miss.Nagh.ma, Miss, tanveer, Miss Shina <Bib% Miss. Zaina6 6i6i, Miss.$figar, Miss.
MataJHra, Miss 9/ififiat Masood^ for their good wishes and encouragement.
My others friend Mr.Aziz, Mr.Sultan Hasan, Mr.AhmadAG, Mr.'Wajid AG, Mr.
Haneef, Mr. Mohd iCyas ,Mr.AsadZedi, Mr. Mohd Shafeeq, Mr.<RaliidK'Kfian, Mr. Mohd
'KfifeeC, Mr. Haider AG, Mr. AzharAG, (Dr. %ajesh %}imar, Mr. Ajay T&mar, Mr. SyedAsad
AG, Mr.Mohd Feeroz, Mr. Anees Ahmad, Mr. 7(fiaGd' Ansari,Mr.Ja6irAG, Mr. Shahid
<Parvez ,Mr.Jamshed,Mr. JaSirAG a(so deserves a few words of thanks.
At the end I would G^e to thanf^ my room partners and friends Mr. Hirdesh %ymar,
MrSuhadAnsari, Mr. AG)^JKjimar Mishra for their cooperation and encouragement.
TinaGy, I owe a deep sense of gratitude to the authorities ofAGgarh MusGm Vniversity,
myAGna mater, for providing me adequate research faciGties.
(mS<E<EM JWMM> TQVW)
DEPARTMENT OF APPLIED MATHEMATICS ZAKIR HUSAIN COLLEGE OF ENGINEERING & TECHNOLOGY FACULTY OF ENGINEERING, ALIGARH MUSLIM UNIVERSITY, ALIGARH - 202002, U.P. INDIA
v
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HffYMfMlAI ^
Applied Mathematics A. M. U. Aligarh
PREFACE
The present thesis embodies the researches carried out at the Aligarh
Muslim University, Aligarh. The thesis comprises of six chapters. Be
sides the introductory I chapter, the next two chapters deal with some
finite integrals involving generalized polynomial sets, and the multivari-
able //-Function with applications and iV-Fractional calculus of gener
alized if-function. Further two chapters concern Fourier series involving
//-function and Integrals and Fourier series involving //-function respec
tively. The last chapter deals with a finite integral involving generalized
Lauricella's function and the //-function with applications.
Chapter I gives a brief survey of some of the earlier and recent works
connected with the present thesis in the field of various historical back
ground ,definition,some identities, elementary properties and special cases
of H-function and H-function expressed in terms of elementary Special
Function and vice-versa and recent developments on the H-function of
several complex variables due to Srivastava et.all, and multivariate / /-
function of several complex variables introduced by Srivastava and Panda
which is an extension of the //-function and the //-function which is a
vi
Applied Mathematics A. M. U. Aligarh
new generalization of the familiar H-function of Fox, introduced by Hus-
sian,and general class of polynomial sets, Operators of fractional integra
tion, Fractional derivatives and hypergeometric function.
C H A P T E R II deals with the evaluation of the three new finite inte
grals involving the product of generalized polynomial sets and the multi-
variable //-function. These integrals are unified in nature and serve as key
formulae from which one can derive as its particular cases, integrals in
volving a large number of simpler special functions and polynomials. For
the sake of illustration, only one particular case of the integrals obtained
has been given which is also new and is of interest. At the end, applica
tion of the main results have been given by inter-connecting them with
the Riemann-Liouville type of fractional integral operator. The results
obtained are basic in nature and are likely to find useful applications in
several fields notably electrical network, probability theory and statistical
mechanics.
C H A P T E R III deals with Nishimoto's JV-fractional differintegral
of the I-function whose argument involves the product of two power func
tions (z — a)~x and (z — b)~^ (A, \± > 0).On account of the general nature
of the main result, TV-fractional differintegral of a large variety of special
VII
Applied Mathematics A. M. U. Aligarh
functions having general argument follows as special cases of the present
findings. Results of Romero, Pricto and Nishimoto [45], Romero, Kalla
and Nishimoto [44] and Gupta, Goyal and Garg [24] which follow as special
cases of the present study.
C H A P T E R IVdeals with an application of an integral involving sine
function,exponential function, the product of Kampe de Feriet function
and the //-function to evaluate three Fourier series.A multiple integral
involving the //-function has also been evaluated to make its application
to derive a multiple exponential Fourier series. Finally, some particular
cases have been discussed.
C H A P T E R Vdeals with the evaluation of an integral involving an
exponential function, sine function, generalized hypergeometric series and
//-function and the same has been employed to evaluate a double integral
and establish Fourier series for the product of generalized hypergeometric
functions. A double Fourier exponential series for the ^-function has also
been derived.
VIII
Applied Mathematics A. M. U. Aligarh
C H A P T E R VI deals with a finite integral involving generalized
Lauricella function and the ^-function . This integral acts as a key
formula from which one can derive many useful particular cases. For the
sake of illustration two particular cases of the main results have been
obtained. At the end application of the main results have been given
by interconnecting them with the Riemann-Liouville type of fractional
integral operator.
IX
CHAPTER I
Applied Mathematics A. M. U. Aligarh
CHAPTER I
INTRODUCTION
1.1 THE H-FUNCTIONS
This Thesis deals with' H-functions, known in the literature as gen
eralized Mellin-Barnes function or generalized G-functions or Fox's H-
functions. All the recent developments on H-functions are given in this
chapter along with key results. Other useful results are given in other
chapter of the thesis. Applications of the results in statistics and other
disciplines as well as functions of matrix argument are discussed.
The H-function is applicable in a number of problems arising in phys
ical sciences, engineering and statistics. The importance of this function
lies in the fact that nearly all the special functions occurring in applied
mathematics and statistics are its special cases.
Besides the functions considered by Boersma (1962). Mittag-Lefler.
generalized Bessel function due to Wright(1935), the generalization of the
hypergeometric functions studied by Fox (1928) and Wright (1935a, 1940)
are all special cases of the H-function. Except the function of Boersma
(1962), the aforesaid functions cannot be obtained as special cases of
the G-function of Meijer (1946); hence a study of the H-function will
cover wider range than the G-function and gives deeper, more general
and more useful results directly applicable in various problems physical
and biological sciences.
Applied Mathematics A. M. U. Aligarh
1.2 DEFINITION OF THE H-FUNCTION :
Of all the integrals which contain gamma functions in their integrands
the most important ones are the so-called Mellin-Barnes integrals. Such
integrals were first introduced by S. Pincherle in 1888, their theory has
been developed by Mellin (1910) where there are references to earlier
works. They were used for a complete integration of the hypergeometric
differential equation by E. W.Barnes (1908). Dixon and Ferrar (1936)
have given the asymptotic expansion of general Mellin-Barnes type inte
grals.
Dixon and Ferrar (1936) have given the asymptotic expansion of gen
eral Mellin-Barnes type integrals.
Functions close to an H-function occur in the study of the solutions
of certain functional equations considered by Bochner (1958) and Chan-
drasekharan and Narasimhan (1962).
In an attempt to unify and extend the existing results on symmetri
cal Fourier kernels, Fox (1961) has defined the H-function in terms of a
general Mellin-Barnes type integral. He has also investigated the most
general Fourier kernel associated with the H-function and obtained the
asymptotic expansions of the kernel for large values of the argument, by
following his earlier method (Fox, 1928).
It is not out of place to mention that symmetrical Fourier kernels are
useful in characterization of probability density functions and in obtaining
the solutions of certain dual integral equations. In this connection one is
referred to the work of Fox (1965, 1965a), Saxena (1967, 1967a),
2
Applied Mathematics A. M. U. Aligarh
Mathai and Saxena (1969a) and Saxeua and Kushwaha (1972, 1972a).
Charles Fox (1961)introduced a more general function which is well known
in the literature as Fox'H- function or H-function. This function is de
fined and represented by means of the following Millin-Bames type of
contour integral. An H-function is defined in terms of a Mellin-Bernes
type integral as follows:
H^[z\ = H%n
where
(&i»ft)i,?
Tjm.n HP,<,
= kk*®**
i -
{ai,ai)i(a2,a2),---,{ap<ap)
(6i,/?i),(62,/32),---,(69,^)
(1.2.1)
i = (—1) ' , z 7 0 and z^ — exp[£Log \z\ = i arg z], (1.2.2)
in which Log \z\ represent the natural logarithm \z\ and arg z is not
necessarily the principal value.An empty product is interpreted as unity.
Here
m .nrfe-M.nrii-a^^) (/>(£) = —\ J- ? .
. n m-bj + PjO. n r^-c^o j=m+\ j=n+l
(1.2.3)
where m, n, p and q are nonnegative integers such that 0 < n < p, 1 <
m < q; ocj(j = 1, • • • ,p) , (3j(j = 1, • • • ,q) are positive numbers; aj(j =
1, • • • ,p) and bj(j = 1, • • •, q) are complex number.
Applied Mathematics A. M. U. Aligarh
aj(bn + v) and (3n(aj - a - I) (1-2.4)
for v, A - 0,1,- • • ;h = l ,---m; j — 1, •• • ,n
L is a contour separating the points
? = (^)'(i = 1 ' ' " ' m : i / = 0 1 " " )
which are the poles of T(bj — /3j£) {j = 1, • • •, m) from the points
(1.2.5)
e = ( 2 L ± p L ) > 0 = i , . . . i n : l / = o.i....) (1.2.6)
which are the poles of T(l — a7 + otjOiJ = 1» * • • >n)
The contour L exists on. account of (1.2.4). These assumptions will be retained throughout.
In the contracted form of the H-function in (1.2.1) will be denoted by one of the following notation:
H{z), H™[z], H%n
The H-function is an analytic function of z and makes sense if the following existence conditions are satisfied.
Case 1 For all z ^ 0 with // > 0
Case 2 For 0 < \z\ < 7_1with 7 = 0
Here
»=EPj-Z <*j and 7 = ft a? ft PJ03 (1-2.7)
4
Applied Mathematics A. M. U. Aligarh
It does not depend on the choice of L. Due to the occurrence of tile factor z* in the integrand or (1.2.1) it is, in general, multiple-valued but one-valued on the Riemann surface of log z.
1.3 SOME IDENTITIES OF THE H-FUNCTION :
This section deals with certain elementary properties of the .//-function. Many authors have given various properties of this function and the work of Braaksma (1964), Gupta (1965), Gupta and Jain (1966, 1968a and 1969),Bajpai (1969a), Lawrynowicz (1969), Anandani (19692, 1969b) and Skibinski (1970) will be discussed here.
The results of this section follow readily from the definition of the //-function (1.2.1) and hence no proofs are given here.
PROPERTY 1.3.1:
The //-function is symmetric in the pairs (a l5Ai), ,(an,An). likewise (an+1, An+1), , (ap, Ap); in (62,B{), , (bm, Bm) and in (frm+l, Bm+l), , (bg, Bg).
PROPERTY 1.3.2 :
If one of the (aj, Aj) (j — 1, , n) is equal to one of the (bj, Bj) (j — m + 1, , q) [or one of the(6j, Bj) (j = 1 m) is equal to one of the (aj, Aj) {j = n + 1, ,p)], then the //"-function reduces to one of the lower order, and p, q and n (or m), decrease by unity.
Thus one gets the following reduction formula:
Tjm,n X
TTin,n—\
— np-l,q-l
(ai,Ai),(a2,A2) , (ap, Ap)
(6i, BO, (62, B2) , (&,_!, Bq_Y), (a i , A0
(02,^2) ,{aP,Ap)
x (1.3.1)
5
Applied Mathematics A. M. U. Aligarh
provided n > 1 and q > m.
P R O P E R T Y 1.3.3
rm,n p,q
X
[ap, Ap)
%, Bq) ,
rrn,m — nq,p
1
T
(l-bq,Bq)
(1 — ap, Ap) (1.3.2)
This is an important property of the //-function because it enables us to Q v
transform an if-function with fi = £ Bj — £ A* > 0 and arg x to one j=i j=i
with // > 0 and arg (l/x) and vice versa.
P R O P E R T Y 1.3.4
Tjm.n X
(ap, Ap)
for, Bq) j
TTTTl.n
= HP,Q xk
(ap, kAp)
(bq, kBq)
where k > 0.
P R O P E R T Y 1.3.5
„<r TTm,n X np,Q
X
K i Ap)
(V B,) J Tjm.n
= HP,Q xk
(op + aAp, Ap)
(6, + <7B9,Bg)
P R O P E R T Y 1.3.6
(1.3.3)
(1.3.4)
£ r jm,n+l
where p < q.
(0,7),(ap,i4p)
(ap,i4p),(r,7)
r r rm+l ,n _ / i \ r r r " i t i , n (ap,Ap), (0,7)
(r, 7), (&„£,) (1.3.5)
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Applied Mathematics A. M. U. Aligarh
P R O P E R T Y 1.3.7
( l - r , 7 ) , ( a p , i 4 p ) ,
(6„fl , ) , ( l ,7) , (1.3.6)
where p < q.
P R O P E R T Y 1.3.8
(A(t,ap),Ap),
(A{t,bq),Bq), (1.3.7)
where £ is a positive integer,
Q P
M = 5Z bJ ~ E % a n r f (A(*> $r), 7r) 1 1
stands for
te^ f*±i7^ (*L±±Z±~\ U ' 7 v ' l * ' 7 v ' ' l t , 7 rJ-
The result (1.3.7) is given by Gupta and Jain (1966) and is called the multiplication formula for //-functions.
_£ Tjm+l,n ( a p , A p ) , ( l - r , 7 )
(1,7), ( ^ 5 , )
m+l,n ( ~\\r u"i+J.,n \ l) np+l,q+l
Trm,n HP,Q z
(ap, Ap)
(V Bq) = pTT)*1-*^ X tfgg tn (z^)<
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Applied Mathematics A. M. U. Aligarh
SPECIAL CASES :
The .//-function, being in a generalized form, contains a vast number of analytic functions as special cases. These analytic functions appear in various problems arising in theoretical and applied branches of mathematics, statistics and engineering sciences.
In the first place, when
A^Bj^l (j = l,--
it reduces to a Meijer's G-function.
,p; h= 1, ,9)
Tjm,n Hp,q
X
K> i ) '
(V i) J X
ap
M (1.3.8)
A detailed account of Meijer's G-function and its applications can be found in the monograph by Mathai and Saxena (1973a).
The G-function itself is a generalization of a number of known special functions listed on pages 53-68 of the monograph by Mathai and Saxena (1973a). Therefore, not listing all those special cases of the //-function which follows from the G-function. However, they list here a few interesting special cases of the //-function which may be useful for the workers of integral transforms, special functions, applied statistics, engineering sciences and perturbation theory.
1,1 TTi.,1
• " 1 , 1
Z J - 1 , 0
- " 0 , 1
(1 - v, 1)
(0,1)
(b,B) = B-lzblB exp{-zllD) (1.3.9)
I » ( l + z)~v = r(i/) iF0(i/; -z) (1.3.10)
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Applied Mathematics A. M. U. Aligarh
7/1,0 •"0,2
2 a+",i\r-^A Jv(z), 2 , _ r v 2
where Jv(z) is the ordinary Bessel function of first kind.
v2 rr2,0 -"0,2
Z
T fl"",ilf^i = r~aza Kv{z), 2 ' J ' \ 2
where Kv(z) is the modified Bessel function of second kind.
(1.3.11)
(1.3.12)
rj-2,0 H\,2
( a - A + 1,1)
(a + ^ + | , l ) , ( a - / i + £,l)
where W A , / ^ ) is a Whittaker function.
= 2a e~z'2 W^(z), (1.3.13)
rrl,2 #2,2
( 1 - A , 1 ) , ( 1 - M )
(0,1), (1 - 1/, 1)
r(A)rpz) r(i/) 2F1(A,/i;i/;-z), (1.3.14)
The following two special cases of an //-function cannot be obtained from a G-function.
Tjl.P (1 ~~ cipi Ap)
(0,1), ( 1 - & „ £ , )
0 0 fir(aj + v ) ^ = £ i r=o ft T(bj + B,-r)
= P4\ \CLp, A.p)\
(bq,Bq); , (1.-3.15)
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Applied Mathematics A. M. U. Aligarh
which is called Maitland's generalized hypergeometric function.
The above series has been studied in detail by Wright (1935a). An interesting particular case of (1.3.17) gives a relation between an //-function and a Maitland's generalized Bessel function Jtfiz). The result is:
rl,0 oo
H$[z |(0,l),(-^/x)]= £ (-z)7
r0 r! r(i + v + H This result (1.3.16) is given by Braaksma (1964, p. 279)
One has also
= J>\z) (1.3.16)
frP,l E(p; ar : q; bs : z), (1.3.17) (1,1), ( M )
(Op, 1)
where E denotes MacRobert's /^-function (MacRobert, 1969).
The following special cases of the //-function occur in the study of certain statistical distributions.
tr2,0 •"2,2
.
Z
(al + (31-l,l)1(a2 + p2-l:l)'
( Q i - l , ! ) , ^ - ! , ! )
7a2-\ (] _ ^ f t + f t - l
r(ft + ft) 2 Fi
( (a2 + ft-<*i,ft); \ l-z\(\z\<l).
{ (ft+ft); ) (1.3.18)
rrl.O ^ 1 , 1
r
z ( a + 1/2,1)'
(M) = -K-1'2 Za (I-Z)-1'2 ( | 2 | < 1 ) .
(1.3.19)
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Applied Mathematics A. M. U. Aligarh
H. 2,0 2,2
(a+1/3 ,1) , (a + 2/3,1)^
(a,l),(a,1) )
: (27T)1/2 za~l - 2za-1/27r-1/2 2Fi(l /2,1/2; 3/2; z)
= p ^ y ^ (1 - *)1/2 2*i(l/2,1/2; 3/2; 1 - z) (\z\ < 1). (1.3.20)
(logarithmic case)
1.4 H-FUNCTION EXPRESSED IN T E R M OF ELEMENTARY SPECIAL FUNCTION :
rl,0 H$ [z\(a,l)] = z«e-z. (1.4.1)
" 1 , 0 -"0,2 z
(M)"
(ft i) J * _ z * ( l _ z r / J - l l w < 1
T{a- (3)
(1.4.2)
& 1,0 0,2
(1/2,1)
( a , l ) ( - a , l ) = n-l/2cos(an)el/2Ia (?
(1.4.3)
rrl ,0 -"0.2
(1/2,1)
(ft!)(-/?,!) -1/2e-1/2/^ g (1.4.4)
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Applied Mathematics A. M. U. Aligarh
H$ [z|(a, 1), (P, 1)] = z^^ja^(2z^). (1.4.5)
H2Qj [z\(a, 1), (0,1)] = 2zla+M2Ka-0{2zV2).
(1.4.6)
1.5 ELEMENTARY SPECIAL F U N C T I O N EXPRESSED IN T E R M OF H-FUNCTION :
1,0 z*e-* = H$ [z\(a,l)} (1.5.1)
sxnz '-KZ\ 1/2
#0,2 4
( \A)
(-il)
1/2rrl,0 * ' #0,2 4
( U)
(o, i) (1.5.2)
cosz = 7T2 1/2
# . 1.0 0,2
EL 4
("il)
( i,i)
1/2rrl.O ^ ' #0,2
z2
4
(0, 1)1
( l . i ) J (1.5.3)
i,o log(l±z) = ± f f $ ± 2 (1,1), (1,1)
(1,1), (0,1) (1.5.4)
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Applied Mathematics A. M. U. Aligarh
( | ,1) , (0,1)'
(0,1), ( i i ) (1.5.5)
1.6 The //-function :
Feynman path integral are reformulation of quantum mechanics and
are more fundamental than the conventional one in terms of operators be
cause in the domain of quantum cosmology the conventional formulation
may be fail but Feynman path integrals still apply [50].Inayat-Hussain[56]
has pointed out the usefulness of Feynman integrals in the study and de
velopment of simple and multiple variable hypergeometric series which in
turn are very useful in statistical mechanics.
Hussain has introduced in another paper [54] the //-function which
is New Generalized of familiar of //-function of Fox[48].The //-function
conations the exact partition function of the Gaussian model in statistical
mechanics function useful in testing hypothesis and several others as its
special cases.
A function more general than well known Fox H-function was intro
duced by Inayat-Hussain ([56], [57]). This function has been put on a
firm footing by Buschman and Srivastava [18].
The //-function occuring in the chapter will be denned and repre
sented in the following manner [18]
log l + z"
= zHiS — 2
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Applied Mathematics A. M. U. Aligarh
H^N[z\ = R»# •P,Q \Q
(aj, a j ; A / ) I , N , (%, <*J)N+I,P
(&i» ft)i,Wi fei A'i BJ)M+\.Q
1 /-too _ x f
Y7T7. J—lOO 27T2 (1.6.1)
where
AT
0(0 = nrft-W.n{r(i- f l i + a ^
n {ra-^ + ftO}^. n r ^ - ^ o (1.6.2)
which contains fractional powers of some of the gamma functions.
Here, and throughout the chapter a,j(j = 1, • • •, P) and bj(j = 1, • • •, Q)
are complex parameters, a;- > 0 (j = 1, • • •, P) , fy > 0 (j = 1, • • •, Q)
(not all zero simultaneously) and the exponents Aj {j = 1, • • • ,N) and
Bj (j — M + 1, • • •, Q) can take on non-integer values.
The contour in (1.6.1) is imaginary axis Re(^) = 0. It is suitably in
dented in order to avoid the singularities of the gamma functions and
to keep those singularities on appropriate sides. Again, for Aj(j =
l,...,iV) not an integer, the poles of the gamma functions of the nu
merator in (1.6.2) are converted to branch points. However, as long as
there is no coincidence of poles from any V(bj — (3j£) (j — 1,..., M) and
r ( l — aj + ctj£) (j — 1,..., TV) pair, the branch cuts can be chosen so that
the path of integration can be distorted in the usual manner.
Evidently, when the exponents Aj and Bj are all positive integers, the
H-unction reduces to the well-known Fox's H-function ([48], [160]).
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Applied Mathematics A. M. U. Aligarh
The basic properties and the following sufficient conditions for the abso
lute convergence of the defining integral for the ^-function have been
given by Buschman and Srivastava [18].
Buschman and Srivastava [18] has proved that the integral on the right
hand side of (1.6.1) is absolutely convergent
when Q, > 0 and | argz |< -nfl, where
M N Q P
n= Ef t + E V i - E Bib- E «i>o. (1-6.3) j '= l j=l j=M+l j=N+l
The behaviour of the //-function for small values of |z| follows easily
from a result recently given by Rathie [113, p.306, eq. (6.9)].
H%$[z\ = 0( |zD, 7 = mm1<y<m [Re{^-)]
Investigations of the convergence conditions, all possible types of con
tours, type of critical points of the integrand of (1.6.1), etc. can be made
by an interested reader by following analogous techniques given in the
well known works of Braaksma [14], Hai and Yakubovich [53]
If we take A,- = 1 (J = 1, • • •, iV), Bj = 1 (j = m + 1 , • • • ,Q) in (1.6.1).
the function HPQ reduces to the Fox's H-function.
1.7 MULTIVARIABLE H-FUNCTION :
Recently, H. M. Srivastava and R. Panda introduced and studied an
extension of the H-function in several complex variables in a series of
papers (175a,) (1976), (1976a,b) (1978), (1979).
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Applied Mathematics A. M. U. Aligarh
The multivariable //-function of several complex variables occuring in
this paper by Srivastava and Panda ([60].see also[64]) is represented by
means of the multiple Mellin-Barnes integrals in the following form;
0,N:mini:---:mr,nT rrr i _ TTU,IV :mlnl:-:mT
n[ZU---,Zr\ - tlRQ:Pl,qi:...;pr,q,
Z\ {ay, a),
4r)) \Cj J lj )l,pi ) > \Cj > 7 j )l,Pr
(d),S}Kqi; M]4])Ur
[Zirujy JL\ JLT
(1.7.1)
where u> = -/— 1
/(D Jf(D (1) , J D ,
^te) n r (d^ - J^ i ) n r(i-cy> + 7j%) =i j = i
;(D x. Jt(D (1) n r(i - d)l) + sy^,) n r(c^-7 je) i = l 7=n 1
, V e { l , - - . , r }
(1.7.2)
<M& •••&•) = n r(i -Uj+z af^i)
j=i j=i
fl , r(a,- - £ a j 1 ^ ) . fl , r ( l - 6j + £ ^ (1)6) i = l j = l
(1.7.3)
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Applied Mathematics A. M. U. Aligarh
The above notation slightly differs from the notation used by Srivastava
and Panda is consistent with the notations used for the H-functions of
one and two variables in this chapter.
In (1.7.1), i in the superscript (i) stands for the number of primes,
e.g.,6^) = b', M2) = 6", and so on; and an empty product is interpreted as
unity.
Suppose, as usual, that the parameters
a-jj = 1, • • - ,? ; cf,j = 1, ••• ,?*;
bjj = 1, ••• ,? ; df,j = l,---,<fc;Ve { l , - - - , r}
are complex numbers, and the associated coefficients
r (0 • 1 (0 • 1
<
. / ? f , j = l , -- . ,g; iSi),j = l , - , « ; V e { l 1 - 1 r }
are positive real numbers such that
A = E ^ + E if ~ E 0J° - E #> < o (1.7.4) j=i j = i j = i j = l
and
^ = - E 4 } + E 7J° - E 7J0 - E P? + E 4l) - E sf] > o (1.7.5) j=i j = i j = i j = i j = i j = i
V e { l , - , r }
where the integers n,p,q,mi,rii,pi and <& are constrained by the in
equalities 0 < n < p, q < 0,1 < mi < q{ and 0 < ra^ < pjVi G {1, • • •, r}
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Applied Mathematics A. M. U. Aligarh
and the equalities in (1.7.4) hold for suitably restricted values of the
complex variables z\, • • • ,2,...The sequences of parameters in (1.7.1) are
such that none of. the poles of the integrand coincide, that is, the poles
of the integrand in (1.7.1) are simple. The contour Lj'm the complex
£i-plane is of the Mellin.Barnes type which runs from -o>oo to UJOO with
indentations, if necessary,to ensure that all the poles of T(dj — #j &), j =
1, ....m.j are separated from those of T(l — Cj — Tjj = 1,..., rij, and
r f l - a i + E a f C j ] , J = l , - " , n . V G { l , - - - , r }
It is known that the multiple Mellin-Barnes contour integral represent
ing the multivariable H-function (1.7.1) converges absolutely under the
conditions(1.7.1) [Srivastava and Panda 156, p. 130, Eq. (1.3)], when
| a r ^ ) | < i n , 7 r , V € { l , - - - , r } (1.7.6)
the points Zi = 0, i = 1, • • •, r, and various exceptional parameter values,
being tacitly excluded.
Furthermore, we may recall the known asymptotic expansions [Srivastava
and Panda 156, p. 131, Eq. (1. 9)] in the following convenient from:
f o(\z1\a*->'(\zr\°*,max(\z1\---(\zr\->0
H{Zl,---,zr} = I (1.7.7) ( oflzil"1 • • • {\zr\
Pr, max{\Zl\ • • • {\zr\ -> 00
where,with i = , ••• , r
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Applied Mathematics A. M. U. Aligarh
(0 /jf(0> a* = m m { ^ e ( 4 l V ^ 0 } i = i, m,: (1.7.8)
ft = min{Re{cf - l/lf)} j = 1, • • • n< ,
provided that each of the inequalities in (1.7.4), (1.7.5) and (1.7.6) holds.
If af = ... = af\j = lr-,Pi;Pf] = ••• = pf\j = 1, •••,<& in
(1.7.1), one gets a special multivariable H-function studied by R.K. Sax-
ena [(1974), (1977)]. On the.other hand, if all of the Greek letters are
chosen to be one, the H-function of several variables defined by (1.7. 1)
reduces to the corresponding G-function of several variables studied by
Khadia and Goyal (1970).
Moreover, in terms of the generalized Lauricella function which was de
fined earlier by Srivastava and Daoust [(1969a), p.454], find it worthwhile
to recall here the known relationship [156]:
rr0,p:l,pi ; - ; l , p r
p,g:pi,9i+i;-;Pr,9r+i
Z\
£***
(o7-; a),.., ajr))i,p : (cj, j})hpi;..; (cjr), j \ r ) ) h P r
(bjjj, . . , / ? ] % : (o, l)(d},S})Wi..; (0, l)(4\S^\qr
nr(i-aj)nr(i-c;)...nr(i-cf))
ftr(i-yfir(i-^...fir(i-d<r») ™;* j = l j = l j = l
/ ( l - a , - ; a ) , » > j > ^ j •
v (i-M)>
, ^ ) I , P : ( 1 - ^ T ; ) I , P I ; ;(l-4\^)hpr; \
4%„ :(l-d},(Jj)lift; ;(l-4r ) ,^%r; ' ' / (1.7.9)
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Applied Mathematics • • A. M. U. Aligarh
Many interesting general expansion theorems have been established for
the H-function of several variables by Srivastava and Panda in their well-
known papers (1975a, 1976, 1976b). Srivastava and Panda [155] have also
given multivariate generating relations whose two-variable analogues are
reproduced [155]. The same authors (1978) have introduced two gen
eral classes of multiple integral transformations whose kernels involve
the H-function of several variables; these multidimensional integral trans
forms provide interesting unifications (and extensions) of various classes
of known integral transformations. Certain useful interconnections be
tween these transforms and several other known integral transforms are
also given and many properties are studied ,by them. Two interesting
multiple integral transformations of the H-function of several variables
have been given in another paper by Srivastava and Panda (1979) in
which they indicate several possible applications of the operational tech
niques provided by their results. R. K. Agrawal (1980c), S. P. Goyal and
R.K. Agrawal (1980b, 1981) and others have studied certain statistical
distributions involving the H-function of several variables. Applications
of the multivariable H-function in heat conduction are given in the work
of Prasad and Singh (1977), Siddiqui(1978) and others.
1.8. ORTHOGONAL POLYNOMIALS: Orthogonal polynomials con
stitute an important class of special functions in general and hypergeo-
metric functions in particular. The subject of orthogonal polynomials is
a classical one whose origins can be traced to Legendre's work on plan
etary motion with important applications to physics and to probability
and statistics and other branches of mathematics,
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Applied Mathematics A. M. U. Aligarh
the subject flourished through the first third of this century. Perhaps as
a secondary effect of the computer revolution and the heightened activity
in approximation theory and numerical analysis, interest in orthogonal
polynomials has revived in recent years.
The ordinary hypergeometric functions have been the subject of ex
tensive researches by a number of eminent mathematicians. These func
tions plays a pivotal role in mathematical analysis, physics, engineering
and allied sciences. Most of the special functions, which have various
physical and technical applications and which are closely connected with
orthogonal polynomial and problems of mechanical quadrature, can be
expressed in terms of generalized hypergeometric functions. However,
these functions suffer from shortcoming that they do not unify various el
liptic and associated functions. This drawback was overcome by E. Heine
through the definition of a generalized basic hypergeometric series.
1.9. THE CLASSICAL ORTHOGONAL POLYNOMIALS
The hypergeometric representation of classical orthogonal polynomial
such as Jacobi polynomial, Gagenbauer polynomial, Legendre polyno
mial, Hermite polynomial and Laguerre polynomial and their orthogonal
ity properties, Rodrigues formula, recurrence relation and the differential
equation satisfied by them are given in detail in Rainville [122], Lebedev
,Szego, Carlson . Mention few of them:
Jacobi Polynomial
The Jacobi Polynomials P^a'^(x) are defined by generating relation
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Applied Mathematics A. M. U. Aligarh
£ P^0)(x)tn = [1 + l/2(x + l)t]°[l + l /2( i - l)t]^ (1.9.1) n=0
Re(a) > - 1 , Re(0) > - 1
The Jacobi Polynomials have a number of finite series representation one
of them is given below:
pW)(r)t - ^ (l + a)n(l + a + ^)n+fc (x-l\k
n { ) n ~ hk\(n-k)\(l + a)k(l+a + 0)n { 2 ) ' (1.9.2)
For j3 = a the Jacobi Polynomial P°'a(x) is known as ultra spherical
polynomial
which is connected with the Gegenbauer polynomial C^\x) by the rela
tion
pM{x) _ (L+«)nQ + 1 / 2 (*) ( 1 9 3 )
^ W - ( I + 2Q)B " [ y-dj
For a = /? = 0, equation (1.9.2) reduces to Legendre Polynomial Pn(x).
Hermit e Polynomial
Hermite Polynomial are defined by means of generating relation
oo ^2
exp(2x£ - t2) = £ Hn(x) —, (1.9.4)
valid for all finite x and t and we can easily obtained
K2)(-l) f cn!(2x)n-2 f c
H"{X) = £ * ! ( » - 2 * ) ! • (L9'5)
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Applied Mathematics A. M. U. Aligarh
Associated Laguerre Polynomial
The associated Laguerre Polynomial L^\x) are defined by means of
generating relation.
E L^(x)tn = (1 - *)-("+1)exp — . (1.9.6) n=0 \ t — 1 /
For a = 0, the above equation (1.9.6) yield the generating function
for simple Laguerre Polynomial Ln(x).
A series representation of L^(x) for non negative integers n, is given
by
(a) = A ( - l )* (n + a ) ! s* L " W £bfc!(n-fc)! ( a + *:)!• ( }
for o; = 0, equation (1.9.7) gives the definition of Laguerre polynomial.
Laguerre Polynomial L^\x) is also the limiting case of Jacobi Poly
nomial
Hypergeomertic representations
Some of the orthogonal polynomials and their connections with hy-
pergeometric function used in our work are given below:
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Applied Mathematics A. M. U. Aligarh
Jacobi Polynomial
/><«•«(,) = f + n U
Gagenbauer Polynomial
«w-(B+rv> Legendre Polynomial
Pn(z) = P<?>%) = 2F,
-n, a + (3 + n+ 1 ;
a+ 1
1 - 2
-n, 27 + n ;
7 + 1/2 ;
1 - 2
—n,n + 1 1 - 2
^M r ( i - / x ) U - i /
Hermite Polynomial
*+1Y'\Fi
•n, n + \ : 1 - 2
H„(z) = (22)n2F0
-71 1 n 1 n n i —) Z
- 2
2 '2 2
Laguerre Polynomial
L(a){z) = ( i±^ l F l [_n . 1 + a . 2]
(1.9.9)
(1.9.10)
(1.9.11)
(1.9.12)
(1.9.13)
(1.9.14)
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Applied Mathematics A. M. U. Aligarh
Other familiar generalization (and unification) of the various poly
nomial are studied by Srivastava and Singhal , Srivastava and Joshi .
Srivastava and Panda , Srivastava and Pathan , and Shahabuddin .
1.10. HISTORICAL DEVELOPMENTS OF FRACTIONAL
CALCULUS
The fractional calculus, like many other mathematical disciplines and
ideas, has its origin in the striving for extension of meaning. Well known
examples are the extensions of the integers to the rational numbers, of
the real numbers to the complex numbers, of the factorials of integers to
the notation of the g-function. In differential and integral calculus the
question of extension of meaning is : Can the derivatives -^ of integer
order, n > 0, respectively the n-foid integrals, be extended when n is any
number fractional, irrational or complex ? The affirmative answer has led
to the so-called fractional calculus, a misnomer for the theory of operators
of integration of differentiation of arbitrary (fractional) order and their
applications.
Thus the theory of fractional calculus is concerned with the nth
derivative and n-fold integrals when n becomes an arbitrary parameter.
One versed in the calculus finds that J^p- is nothing but an indefinite
integral in disguise. But fractional orders of differentiation are more mys
terious because they have no obvious geometric interpretation along the
lines of the customary introduction to derivatives and integrals as slopes
and areas. If one is prepared to dispense with a pictorial representation.
however, will soon find that fractional order derivatives and integrals are
just as tangible as those of integer order and that a new dimension in
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Applied Mathematics A. M. U. Aligarh
mathematics opens to him when n of the operator ^ becomes an arbi
trary parameter. It is not a sterile exercise in pure mathematics, many
problems in the physical science can be expressed and solved succinctly
by recourse to fractional calculus.
This is a discipline, "as old a mathematical analysis. It can be cate
gorized, briefly, as applicable mathematical analysis. The properties and
theory of fractional operators are proper objects of study in their own
right. In recent decades, they have been found useful in various fields :
theology, quantitative biology, electrochemistry, scattering theory, diffu
sion, transport theory, probability, statistics, potential theory and elas
ticity etc. However, many mathematicians and scientists are unfamiliar
with this topic and thus, while the theory has developed, its use has
lagged behind ".
These reasons made Prof. B. Ross, the author of the above cited
phrase, backed by other famous fractional analysts : Profs. A. Erdelyi, I.
Sneddon and A. Zygmund to organize in 1974 in New Haven the histori
cal first international conference on fractional calculus. Ten years later, a
second international conference on fractional calculus took place in Scot
land, at Ross Priory - the Strathclyde University's stunning property on
the shores of Loch Lomond. Later on, in 1989, the third international
conference was organized in Tokyo, on the occasion of the hundredth
anniversary of Nihon University.
Going back to the history, it seems that Leibnitz was first to try to
extend the meaning of a derivative -^ of integer order n.
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Perhaps, it was nave toying with the symbols that prompted L' Hospital
to ask : "What of n be | ? " Leibnitz, in 1695, replied, "It will lead to
a paradox" but added prophetically. "Prom this apparent paradox, one
day useful consequences will be drawn". In 1819 the first mentioning of a
derivative of arbitrary order appeared in a published article by Lacroix.
Later on, Euler and Fourier gave meaning to derivatives of arbitrary order
but still without examples and applications. So, the honour of devising
the first application belonged to Abel in 1823. He applied fractional cal
culus techniques to the solution of an integral equation, related to the
so-called isochrone (tautochrone) problem. Abel's solution was so ele
gant that it probably attracted the attention of Liouville to make the
first major attempt in giving a logical definition of the fractional deriva
tive, starting from two different points of view. In 1847, Riemann, while a
student, wrote a paper which was eventually published posthumously and
gave the definition, whose modifications is now known as the Riemann-
Liouville integral. Like other mathematical ideas, the development of
fractional calculus has passed through various errors, absurdities, con
troversies, etc. That sometimes made mathematicians distrusting in the
general concept of fractional operators. Thus, it has taken 279 years, since
L' Hospital first raised the question, for a text to appear entirely devoted
to the topic of fractional calculus: the book of Oldham and Spanier .
The intermediate period (1695 - 1974) is thoroughly described by Ross in
his "chronological Bibliography of the Fractional Calculus with commen
tary" . Among the many authors contributing to the topic the names of
Lagrange, Laplace, De Morgan, Letnikov, Lourant, Hadamard, Heaviside,
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Hardy, Littlewood, Weyl, Post, Erdelyi, Kober, Widder, Osier, Sneddon,
Mikolas, Al-Bassam can be mentioned. A more detailed exposition could
be found in the historical remarks of Ross and Mikolas as well as in the
surveys of Sneddon and Al-Bassam and the huge encyclopaedic book of
Samko, Kilbas and Marichev .
1.11. OPERATORS OF FRACTIONAL INTEGRATION
Certain integral equations can be deduced from or transformed to
differential equations'. In order to make the transformation the following
Leibnitz's rule, concerning the differentiation of an integral involving a
parameter is very useful.
i I F{*,W = J±F{Xit)« + F{x,B)*JL-F{x,A)u , uu , A{x A
The formula (1.11.1) is valid if both F and | £ are continuous functions
of both x and t and if both A'(x) and B'(x) are continuous.
A useful application of the formula (1.11.1) is the derivation of the
following formula:
Jj---jf(x)dxdxr-dx^j~^—f(x-t)n-lf(t)dt (1.11.2) a a a n times ^ '' a
n times
Formula (1.11.2) is known as Cauchy's Multiple Integral formula. It
reduces n-fold integrals into a single integral. Fractional integration is an
immediate generalization of repeated integration.
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If the function f(x) is integrable in any interval (0, a) where a > 0, the
first integral Fi(x) of f(x) is defined by the formula
X
Ffc) = J f{t)dt 0
and the subsequent integrals by the recursion formula
X
Fr+l{x) = J Fr{t)dt r = l ,2 ,3 , - - -a
It can easily be proved by induction that for any positive integer n
1 x
Fn+i{x) = - J{x - t)nf{t)dt (1.11.3)
n. Q
Similarly an indefinite integral Fn(x) is defined by the formulae
OO CX)
Fi{x) = -Jf{t)dt, Fr+l{x) = ~JFr{t)dt, r = 1,2,... X X
and again it can be shown by induction that for any positive integer n.
1 °° Fn+i{x) = -f{t-x)nf{t)dt. (1.11.4)
n- x
The Riemann-Liouville fractional integral is a generalization of the
integral on the right-hand side of equation (1.11.3). The integral
Ra {f(t);x} = fL^j(x- tr-lf(t)dt. (i.n.5)
is convergent for a wide class of function f(t) if fie a > 0. Integral
(1.11.5) is called Riemann-Liouville fractional integral of order a.
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Integrals of this kind occur in the solution of ordinary differential equa
tions where they are called Euler transforms of the first kind. There
are alternative notations for Ra {f(t);x} such as Iaf{x) used by Marcel
Riesz.
The Weyl fractional integral is a generalization of the integral on the
right-hand side of equation (1.11.4). It is defined by the equation (Weyl)
Wa{f(t);x} = ^-jJ(t-x)a-1f(t)dt, Re a > 0. (1.11.6)
A pair of operators of fractional integration of a general kind have
been introduced by Erdelyi and Kober and it seems appropriate to call
the operators Iv,a, K^a defined below which are simple modifications
of these operators, the Erdelyi-Kober operators. The operators IVA and
K^a are defined by the formulae
V f(x) = x-2*-2* Ra {p f (rf) ; x2)
s KTT / (z2-^)"1 «¥V(# (1.11.7) r ( a ) J
and
K„,« /M = J*1 Wa {t-°-i f (ti) ; x2}
j g J (u2 - x2)"-1 , , - - * « /(„) du (l.u.8)
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where Re a > 0 and Re rj > — | .
The modified operator of the Hankel transform is defined by
- / tl~a J»,+t,(xt) f(t) dt. (1.11.9) x) 0
1.12. FRACTIONAL DERIVATIVES AND HYPERGEOMET-
RIC FUNCTIONS:
The simplest approach to a definition of a fractional derivatives com
mences with the formula
^-{eaz) = Da7{eaz) = aa{eaz) (1.12.1)
dza
where a is an arbitrary (real or complex) number.
For a function expressible as
00
/(*) =E Cn exp(an, z) (1.12.2) n=0
Liouville defines the fractional derivative of order a by 00
Daz {f(z)} = E <Cn exp (anz) (1.12.3)
n=0
In 1731 Euler extended the derivative formula
Dn2 {(zx)} = A(A - 1) (X~n + l)zx-n (n = 0,1, 2. • • •)
T(A + 1) A „ V ' zx~n (1.12.4) T(A-n + l)
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to the general form :
where \i is an arbitrary complex number.
Another approach to fractional calculus begins with Cauchy's iterated
integral :
D'zn {/(*)} = / / fff(h)dtldt2---dtn (1.12.6)
0 0 0 0
writing \x for —n, we get
D~n { / « } = f ^ y / / ( * ) ( * " t)-"-ldt, Re (fi) < 0, (1.12.7)
where the path of integration is along a line from 0 to z in the complex
t-plane and Re (/i) < 0.
Equation (1.12.7) defines the Riemann-Liouville fractional integral of or
der —/i; it is usually denoted by I~^f(z).
In case m — 1 < Re (fi) < m (m — 1, 2, 3, ), it is customary to
write (1.12.7) in the form:
dm
D" {I{Z)} = d^ D"~m {I{Z)}
dT
dxm T(-fi + m) l—-jf{t)(z - t)-^m-'dt. (1.12.8)
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The representation (1.12.7) is consistent with (1.12.5) when f(z) — zx.
since
Dt { ^ = ^ 1 ^ - ^ ^ r(-/i)
_A-/x
= _ B ( A + l , - „ )
r(A + 1) zX_^ R ^ > _ ^ R ^ < Q (1.12.9) 7(A-M + 1)
starting from the iterated integral
A"" {/(*)} = / / Jjf(ti)dtldt2---dtn Z tn *! <2
1 oo
= 7 7TT / /(*) (* - ^ ) n _ 1 * n = 1,2, • • • (1.12.10) [n - l j !
and replacing nby -fi, we are led to the definitions
1 °° oo0? {/(*)} = fjZ^jj / / W (* - 2 ) _ / X _ 1 d i ' MM) < 0, (1.12.11)
°°Di {f{Z)} = d ^ ( ° ° D r m { / ( Z ) } )
<f
dzr L V. J fit) (t-z)-"-™-1 dt) (1.12.12)
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m — 1 < Re(fi) < m (m = 1, 2, 3, ).
Equation (1.12.11) defines the Weyl fractional integral or order -/i; it
is usually denoted by K~(l f{z).
There is yet another approach based upon the generalization of Cauchy's
integral formula:
D': {/(*)} = ^<f> , ^ , dt, (1.12.13)
which is employed by Nekrossov.
The literature contains many examples of the use of fractional deriva
tives in the theory of hypergeometric functions, in solving ordinary and
partial differential equations and integral equations, as well as in other
contexts. Although other methods of solution are usually available, the
fractional derivative approach to these problems often suggests methods
that are not so obvious in a classical formulation.
Fractional derivative operator plays the role of augmenting param
eters in the hypergeometric functions involved. Applying this operator
on identities involving infinite series a large number of generating func
tions for a variety of special functions have been obtained by numerous
mathematicians.
The following theorem on term-by-term fractional differentiation em
bodies, in an explicit form, the definition of a fractional derivative of an
analytic function:
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T H E O R E M 1. If a function f(z), analytic in the disc \z\ < p, has the
power series expansion
00
f{z) = E *n z\ \z\ < p, (1.12.14) n=0
oo
r ( A - / / ) n=0 ( A - / J ) „
provided that fie (A) > 0, fie (/z) < 0 and \z\ < p.
Yet another theorem is stated as follows:
T H E O R E M 2. Under the hypotheses surrounding equation (1.12.14),
oo
D>: {zx~l f(z)} = E <*n D? {2A+n-1} n=0
r (A- /x) R=0 (A- /z) n
provided that fie (A) > 0 and \z\ < p.
The following fractional derivative formulas are useful in deriving gen
erating functions :
= p|^y *"_1 4 3 ) [A, a, 0, 7; /i; az, bz, cz] (1.12.17)
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Re (A) > 0, \az\ < 1, \bz\ < 1, \cz\ < 1 ;
nA-/x ( , A - 1 tf-l(l-v)-a2Fi
a,/?;
7;
x i-y
= ^TT ^ F2 [a, £ A; 7, K x, y],
Re (/) > 0, |x| + |y| < 1.
In particular (1.12.17) with c = 0 yields
(1.12.18)
Dxz-'
1 {zx~\l - az)-a (1 - bz)-0}
r(A) s l Fl [A, a, /?, 7; /i; az, &z], (1.12.19)
He (/) > 0, \az\ < 1, |6z| < 1.
on the other hand, in its special case when a = 1 and 6 = c = 0, (1.12.17)
reduces immediately to
He (A) > 0, \z\ < 1.
A, a;
/^;
(1.12.20)
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1.13. N-FRACTIONAL DIFFERINTEGRAL : The iV-fractional
differintegral a function given by Nishimoto [114] is defined as follow; Let
D = {£)_,£>+}, C = {C_, C+}, C- be a curve along the cut joining two
points z and - c o + UJ Im(z),
C+ be a curve along the cut joining two points z and oo + u Im(z),
D_ be a domain surrounded by C_, D+ be a domain surrounded by C+.
(Here D contains the point over the curve C).
Moreover, let / = f(z) be a regular function in D, then the fractional
differintegral of arbitrary order v G R (derivatives of order v for v > 0
and integrals of order v for v < 0) of the function / (z) , if | (/)„ | exists,
is defined by
(/)_„ = lim (/)„ (m £ z+) u=—m
where -7r < arg(( - z) < n for C_, 0 < arg(( - z) < 2ir for C+, C # 2
,and T is The Gamma function.
1.14 THE GENERAL CLASS OF POLYNOMIAL SETS:
Srivastava introduced the general class of polynomial[163]
S Q N = E ^ AQaxQ,n = 0,1,2,- •• (1.14.1)
cr,=0 <*!
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Applied Mathematics A. M. U. Aligarh
Where P is arbitary positive integer and coefficientSi4[Q,a(Q,a > 0) are
arbitrary constant,real or complex.
The generalized polynomials (multivariable) defined by Srivastava [163]
in the following manner;
A[Q„<»i, ••-,«*<»*] * ? , • • • , * ? - (1.14.2)
where Q^ Pj ( j = 1, • • •, R ) are non - zero arbitrary positive inte
gers. The coefficients A[Qi,a\, • • •, QRC(R} being arbitrary constant,real
or complex.
If we take R = 1 in the equation (1.14.2) and denote A [Q\, ct\\ thus ob
tained by AQU Ql, we arrive at the well known general class of polynomials
SQ [X] introduced by Srivastava [147].
The generalized polynomial set defined by Raizada [123,p.64,eq.(2.1.2)]
in the following type formula
With the differential operator Tkj defined by
TkJ=xl(k + x—) (1.14.3)
Though the polynomial set S",/3,r [x] can also be derived from the general
sequence of function defined by Agarwal and Chaubey [3] (see also [162].
problem 16,p.447), it unifies and extends a number of classical
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polynomials studied by various research workers such as Kroll and Frink
[95], Gould and Hooper [51], Singh and Srivastava [131], Chatterjea [29]
etc. Morover, it can be expressed in the following series from as given in
[123,p.71,eq.(2.3.4)].
S £ A T [X, r, s, q, A, B, m, k, 1} = Bqnx^m+n\\ - rxr)sn lm+n
m^n^m+n6 ( - ^ ( - ^ ( - ^ ( ^ ( q - qn)i
hhtoh p\6\i\t\{l-a-6)i
(-3 \ (i + k + rt\ ( -rxr \p (Ax\6
[-7-snl{—r-Ln{r^) (-B-) <L14-4> Taking A — 1, B = 0 and r —> 0, we arrive at the following polynomial
set
m+n p
S; A 0 [ z , r , 9 > l > 0 , ro ,M] = xqn+l^m+nU{m+n) £ £ P=O t=o
^^±±^lj0xy (,14,, A general class of multivariable polynomials defined as follows by Srivas
tava and Garg in 1987.
P i + i a , - i + - + P „ a n < P
SQ+U n[~xs+l,- • -,—Xn] = Yl (~Q)p.+ia.+1+-+Pna„
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OCs+1- " n -
Where P8+i, • • •, Pn are arbitrary positive integers.The coefficients
A(Q; a s + i , • • •, an) < (P; a* < 0, i = s + 1, • • •, n) arbitrary constant,real
or complex.
1.15 FOURIER SERIES :
Mishra employed (1.15.1), (1.15.2) and (1.15.3) to establish three
Fourier series for the product of generalized hypergeometric function and
also evaluated a double integral and double Fourier exponential series for
Fox's .^-function.
7r , m — n ; / • « * » - « > * & = * • ' " = " ' (1.15.1)
Jo 0 , m f= n ; v '
/•7T
/ eimx cos nxdx =^ < 7r , m = n = 0 ; (1.15.2)
0 , m = n;
/•7T . , z— , ra = n ; / ewlxsin nxdx={ 2 ' ' 1.15.3
^ I 0 , m / n ; Provided either both m and n are odd or both m and n are even
integers.
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1.16.THE GAMMA FUNCTION AND RELATED FUNCTIONS:
The Gamma function has several equivalent definitions, most of which
are due to Euler,
T(z) = J tz~le-ldt, R e ( z ) > 0 (1.16.1)
upon integrating by part, equation (1.16.1) yields the recurrence relation
T(z + 1) = zT{z). (1.16.2)
The relation (1.16.2) yields the useful result
r (n + l)) = n!, n = 0,1,2,---
which shows that gamma function is the generalization of factorial func
tion
The Beta function
We define the beta function B(p,q) by
B(p,q) = Jl^~l{\-x)q-1 dx, Re(p) > 0, Re{q) > 0 (1.16.3)
Gamma function and Beta function are related by the following rela
tion
B(P,q) = Y^T^1 P, 9 ^ 0 , - 1 , - 2 , - . . (1.16.4)
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The Pochhammer symbol
The Pochhammer symbol (A)n is defined by
(\\ — i * , if ?i = 0 i A , n " A ( A + l ) - ( A + n - l ) , if n = l , 2 ,3 , - - . ( 1 1 6 ' 5 )
In terms of Gamma function, we have
(A)„ = ^ p ; A ^ 0 , - 1 , - 2 , - . . (1.16.6)
(A)TO+n — (A)m(A + m)n (1.16.7)
(A)-n - J ^ , n=l,2,3,--. , A^0,±l,±2, (1.16.8)
(—l)m(X) (A)n-m = , : ; V ; " , 0<m<n. (1.16.9)
^1 — A — n)j lm
For A = 1, equation (1.16.9) reduces to
( — l)mn\
(n-m)\ = \ \ , 0 < r a < n . (1.16.10)
Another useful relation of Pochhamer symbol (A)n is included in
Gauss's multiplication theorem:
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Applied Mathematics A. M. U. Aligarh
™ /x + j - r m /
Wrnn = N ™ H ^ m , n = 0,1,2,
(1.16.11)
where m is positive integer.
For m = 2 the equation (1.16.11) reduces to Legender's duplication
formula
(A)2n = 22n{^)n(^ + l)n' 1 = 0,1,2,- . . (1.16.12)
In particular, we have
(2n)! = 2 2 n Q ) n! and (2n-f 1)! = 22n ( J ) n\ (1.16.13)
The Error Function
The error function erf(z) is defined for any complex z by
erf(z) = -= JQZ exp(-£2) dt (1.16.14)
and its complement by
erfc(z) = l - e r f ( z ) = —= T exp{~t2) dt. (1.16.15)
Note that
erf(0) = 0 , erfc(0) = 1 erf(oo) = 1 , erfc(oo) = 0 (1.16.16)
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Applied Mathematics A. M. U. Aligarh
1.17. GAUSSIAN HYPERGEOMETRIC FUNCTION AND GEN
ERALIZATION:
The second order linear differential equation
d w dw z(l-z)-— + [c-{a + b + l)z] — -abw = 0 (1.17.1)
has a solution
^ (a)n(6)„ zn
n=o {c)n n\
where a,b,c are parameters independent of z for c neither zero nor a
negative integer and is denoted by 2-F\(a, 6; c; z) i.e.
2Fl(a^c;z) = £ ^ M ^ ^ (1-17.2) n=0 (c)n n\
which is known as hypergeometric function. The special case a = c, b = 1
or 6 = c, a = 1 yields the elementary geometric series £ —, hence the n=on\
term hypergeometric.
If either of the parameter a or b is negative integer, then in this case,
equation (1.17.2.) reduces to hypergeometric polynomials.
Generalized Hypergeometric Function
The hypergeometric function defined in equation (1.17.2) can be gen
eralized in an obvious way.
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a 1 , a 2 , - - - , a p ; z
fiiifoi • • • ,Pq ; = £
(a:1)n---(ap)n zn
n=0 (#l)n • • • ( / y n n!
= pFq(ai,---ap; Pi,---,%; z), (1.17.3)
where p, 9 are positive integer or zero. The numerator parameter ot\, • • • ap
and the denominator parameter /3i,---/?9 take on complex values, pro
vided that
Pj ^ 0 , - 1 , - 2 , . - - , j = 1,2.--.,9
Convergence of pFq: The series pFg
(i) converges for all | z | < 00 if p < q
(ii) converges for | z |< 1 if p = g + 1 and
(iii) diverges for all z, z ^ 0 if p > q + 1
Further more if we set
u = R* (£&-!>; ) >o, the pi^ series with p = q + 1 is
(i) Absolutely convergent for | z j = 1 if Re (a;) > 0
(ii) Conditionally convergent for | z |= 1, 2 7 1 if — 1 < Re(u;) < 0
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Applied Mathematics A. M. U. Aligarh
(iii) Divergent for | z |= 1 if Re(u;) < - 1 .
An important special case when p = q = 1, the equation (1.17.3)
reduces to the confluent hypergeometric series 1F1 named as Kummers
series , (see also Slater ) and is given by
00 (a) zn
if.(ai <**) = £ g f Z~v (1.17.4)
When p = 2, <? = 1, equation (1.17.3) reduces to ordinary hyperge-
ometirc function 2F1 of second order given by (1.17.2).
1.18. HYPERGEOMETRIC FUNCTION OF TWO A N D SEV
ERAL VARIABLES:
Appell Function
In 1880, Appell [8] introduced four hypergeometric series which are
generalization of Gauss hypergeometric function 2-F1 and are given below:
0 0
Ffab.V; <r, x,y\ = E M'-W-W" *"»" m,n=0 (c)m+n m\ Tl\
(1.18.1)
(max{ |x | , | J / | } < 1 )
0 0 C, r L U ' 1 V* ( a ) m + n ( 6 ) m ( 6 % Xmyn
F2[a,b,b; c,c ; x,y] = X, ,n=o (c)m (cOn m! n! (1.18.2)
(M + M ) < i
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Applied Mathematics A. M. U. Aligarh
P f M , , tll f (a)m(a')n(b)m(b')n xmyn
F3[a,a,b,b; c; x,y] = £ rT - j - y m,n=0 l,C;m_|_n 771! 77,!
(max{ |x | , | j / | } < 1)
rn r 7 / 1 ^ ( a ) m + n ( ^ ) m + n £ m 7 / n
F4[a,b; c,c; x,y\ = ^
(1.18.3)
m o (c)m (c% m! n! (1.18.4)
(7M + >/r7l<l). The standard work on the theory of Appell series is the monogrph by
Appell and Kampe de Feriet [8].
Humbert Function
In 1920, Humbert has studies seven confluent form of the four Appell
functions and denoted by $i , $2 , $3 , # i , # i , Eu E2 four of them are given
below :
*,(«,/* 7; x,y\ = g ^ f ^ ^ g , (U8-5) m.n.=0 I 7 j m + n i*1- «*:
( | X | < 1 , | J/ | < 00)
<i>2[/?,/?'; 7; x,y] = £ ^ 2 % f ^ , ( L 1 8 . 6 ) i,n=0 ( 7 ) m + n m ! 7l! V ' m,:
(| x |< 00, j y |< 00)
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Applied Mathematics A. M. U. Aligarh
$3[/?; 7; x,y] = £ Wm xmyn
,n=0 ( 7 ) m + n ^ ! f l ! '
(| X |< OO, J y |< OO)
(1.18.7)
*I[Q,/3; 7,7'; z,y] = E ( a ) m + n ( j 9 ) m Z m y B
,n=0 ( 7 ) m ( 7 ' ) n ™! n ! '
( | x | < l , | y | < o o )
(1.18.8)
#2[a; 7,V; x,y\ = £ (a ) m + n xmi/ ra„,n
,n=0 ( 7 ) m ( Y ) n TTl! Tl!'
( | X | < OO, I 3/ | < OO).
(1.18.9)
Lauricella Function
In 1893, Lauricella generalized the four Appell functions Fi, F2, F3, F4
to functions of n variables defined and represented as
F£n,[a,&i,..,6„; ci,..,cn; Xi, ..,£„] = i 7 ^ ^ 1 a:&r,---;&n ;
•^1> " " ' j 2^n
0 : C!; - - - ;c„ i
Y^ (^Jmi4—•+mny^l)mi ' ' " \pn)mn %i
mi—mn=0 (Cl )„u • • * (Cn)mn m i ! mn\ (1.18.10)
(|xi ! + • • • + I x n | < l ) ,
FB [ai, ..,an,6i, ..,6n;c;xi, ..,a;n] = F1.'0;'....'0
- : a i ^ i ; • • • \an,bn;
•Eli 1 2-n
-1 1 1
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Applied Mathematics A. M. U. Aligarh
E m\--mn—0
( a i ) m x • • • MmnMm, ' ' • {bn)mn x"1 • • • X™"
(c)mi+...+m„ mx\---mn\
(max{| xi | , - - - , | xn |} < 1),
(1.18.11)
i ( n F c [a, 6; ci,---,cn; x\,- - • ,xn\ p2:0;-;0
a, 6
X\i ' ' ' j ^*n
c i ; - - - ; c. n j
{a)mi+-+mn{bl)mi+-+rnn ^ 1
mi ,7n2, -,mn=0 ( c l ) m i • • • ( C n ) / ^ ^ l ' mn\ (1.18.12)
{yf\xT\ + --- + s/\x7\<l),
An) Ffr'[a,bi,---,bn; c; Xi,---,xn] p i ; ! ; - ; ! ^ l :0 ; - ;0
a :6i ; --- ; 6n;
*M> i "^n
- J ) — j
oo
= E mi,—mn—0
{0')mi+-+Tnn{bi)mi • • • ( 6 n ) m n XX* • • • X™n
\c)ml+" +mn mi\---mn\
(1.18.13)
(max{| xx | , - - - , | xn |} < 1),
Clearly, we have
^ 4 - ^ 2 , ^ f l -,(2) (2) F3, F£" = F4, F ^ = F l5
and
p ( l ) _ ci(l) _ ci(l) _ p ( ! ) _ P ^ — ^ B - tC ~ bD - 2 ^ 1 -
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Applied Mathematics A. M. U. Aligarh
A summary of Lauricella's work is given by Appell arid Kampe de
Feriet [8].
A unification of Lauricella 14 hypergeometric functions F\, • • •, F14
of three variables , and the additional functions HA,HB,HC was intro
duced by Srivastava who defined a general triple hypergeometric series
F^[x,y,z\.
^(3) (a) •,(b);(b')](b"y,(c)-(c');(c") ;
X XI Z
(e) ••:(»); (»'); (s");W;W;OT ; ' '
= £ ( (a ) )^n + P ( (6 ) ) m + n( (^ ) ) r i + P ( (6 / / ) ) P + m((c) ) m ( (c / ) ) n ( (c / Q) p xm
Vn Z?
^ 0 ( (e))m+n+ P((5))rn+n((5 ' ))n+ P((5"))P +m((/ l ))m((/ i /))n((^))P ™! n! p! ' (1.18.14)
with, as usual, (a) abraviates the array of A-parameters
ai,a2---aA, ((a))m = II (<*;)*. = U y , + ™ . (1.18.15) i=i j=i x [aj)
Confluent form of Lauricella function
$2 a n d ^2 a r e two important confluent forms of Lauricella func
tions are given by
^n)\b ft-c-z, ••• zl - ? ^ - M " . J i " - C mi-Tvn=o {ci)mi+-+mn mil • • • mn\
(1.18.16) and
50
Applied Mathematics A. M. U. Aligarh
^ n ) [a ; ci,---,c„; xi,---,xn]
,(«)
£ Mm! + -+m„ x l
mi—mn=0 l c l j r n i ' ' ' [cn)mn Wi l . • • • 777.,;.
(1.18.17)
In terms of ^ , the multivariable extension of Whittaker's MK,fi,
function was defined by Humbert in the following form:
M 1- . . . T \ _ M+1/2 . . M n + i / 2 exp
- 1 /
— ( x i + ••• + £„)
,(» ^ r [ ^ i + --- + M n - « + n/2; 2//1 + l,---2/in + l; X!,---,a;n]. (1.18.18)
1.19. BESSEL FUNCTION AND HYPER BESSEL FUNCTION
Bessel's equation of order n is
x 2<Py , dy , , 2 , 2 n
+ x — + (ar + n )?/ = 0,
where n is non-negative integer. The series solution of the equation
(1.19.1) is
Ux) = £ ( - l ) r (x /2) 2 r + n
,=0r! r (n + r + l)
the series (1.19.2) converges for all x.
In particular,
(1.19.2)
J-m(z) = \ —cos z and J1/2M = « smz. irz
(1.19.3)
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Applied Mathematics A. M. U. Aligarh
Here Jn{x) as Bessel function of first kind. The generating function
for the Bessel function is given by
exp x ( 1M 2 V t
0 0
= E tnJn(x). (1.19.4)
Bessel function is connected with hypergeometric function by the re
lation
•«*> = # £ j " * _ ; i + ri] X
(1.19.5;
Bessel functions are of most frequent use in the theory of integral
transform. For more discussion of the properties of Bessel function .
Modified Bessel's Function
Bessel's modified equation is
x <A2y , dy
+ x (x~ + n)y 0, dx2 dx
the series solution of the equation (1.19.6) is
(1.19.61
'»(*) = E (x/2) 2r+n
(1.19. r=or\ r (n + r + 1)'
where n is non negative integer.
Here In{x) as modified Bessel function. The function In(x) is related
to Jn{x) in much the same way that the hyperbolic function is related to
trigonometric function, and
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Applied Mathematics A. M. U. Aligarh
In(x) = i nJn{ix).
The generalized Bessel functions (GBF) have been the topic of a recent
study by the authors . This research activity was stimulated by the num
ber of problems in which this type of functions is an essential analytical
tool and by their intrinsic mathematical importance. The GBF have
many properties similar to those of conventional Bessel function (BF).
As far as the application of GBF are concerned they frequently arise in
physical problems of quantum electro dynamics and optics, the emission
of electromagnetic radiation, scattering of laser radiation from free or
weekly bounded electrons .
Hyper Bessel Function
The Hyper Bessel function Jmi„(z) of order 2 is defined by
Jm,n\Z) (z/3) m+n
T(m + 1) T(n + 1) 0^2 .; m + l , n + 1; - I - (1.19.8)
and its generating function is defined by
exp z ( 1 -lx + y 3 V xyj\
oo
= £ xmyn (z/3) m+n
m.i .; m + l , n + 1; ,„tr-oor(m + i)r(n + i)° 2
(1.19.9)
Modified Hyper Bessel Function
The modified Hyper Bessel Function I„hn(z) of order 2 is defined by
3 ,
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Applied Mathematics A. M. U. Aligarh
•Lm,n\Z) ( z /3 ) w + n
0-P2 r ( m + l ) r ( n + l )
and its generating function is defined by
; m + l , n + l; ( - (1.19.10)
exp z ( 1 -\x + y + — 3 V xy = £ xmyn {z/3)m+n
n= — DO r ( m + l ) r ( n + l) 0 r 2 _; m + l , n - f 1;
(1.19.11)
The generating function of Hyper Bessel function Jmy..mn(z) of order
n and its modified case Imi...mn(z) are given by
exp . n + 1
xi H h z n •^1 ' ' * "^n/ -
0 0
= £ H i i - - m n = — 0 0
^ 1 XnnJmymn\Z),
(1.19.12)
where
Jmymn\z) —
and
5Z mj (z/n + iy=l
ml! • • -TTIT,! oK. mi + 1- ••mn + l;-
n+l
n + l) (1.19.13)
exp . n + l
where
z ( 1 Xi-\ hXn +
X\ ' ' ' Xji
0 0
2^ 2:1 • • • xnn lmi...rnn[z)
Tti\---mn=—oo
(1.19.14)
54
- ^ A M / ^ > v
Applied Mathematics >7 <. ^ / A. M. U. Aligarh
T ( \ - (Z/n+l)]" c. m i ! • • • r a j
n+11 _; mi -f-1 • • -ran 4- 1;
.n+l, (1.19.15)
For ? i = l , these functions coincide with the Bessel function.
1.20. MITTAG-LEFFLER'S FUNCTION A N D RELATED FUNC
TIONS
The function
0 0
E0{Z) = E -, a > 0 (1.20.1) k=0T(ak + l)
was introduced by Mittag-Leffler and was investigated systematically by
several other authors . En(z), for a > 0, furnishes important example of
entire functions of any given finite order I/a.
We note that
EY{z) = e2, E2{z2) = coshz, £1 / 2U1 / 2 = 27r1/V*Erfc(-z1/2)
(1.20.2)
where erfc denotes the error function, defined by equation (1.16.15)
Many of the most important properties of Ea(z) follow from the in
tegral representation
1 r r ~ V _, Eo = 7T-. / dt
2m Jc t° - z (1.20.3)
0 0
Applied Mathematics A. M. U. Aligarh
where the path of integration C is a loop which starts and ends at — oo,
and encircles the circular disc | t \<\ z |1//a in the positive sense i.e.
—7r < argt < 7r on c. The following Laplace transform of Ea(ta) was
evaluated by Mittag-Leffler:
/•oo , 1
/o e-fE0{taz) dt = — (1.20.4)
where the region of conversion of integral (1.20.4) contains the unit circle
and is bounded by the line Re(z1/,Q) = 1. Humbert [34] obtained a number
of functional relations satisfied by En(z) with the help of integral (1.20.4).
Feller conjectured and Pollard showed that Ea(—x) is completely
monotonic for x > 0 if 0 < a < 1.
We have
E0{-x) = (1 + x)-1 , Eii-x) = e~x (1.20.5)
The function
oo ?k.
E*A*) = Z p r r T f l T ' <*,P>0 (1-20.6)
has properties very similiar to those of Mittag-LefBer's function.
We have
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Applied Mathematics A. M. U. Aligarh
Ea,i{z) =Ea(z), Eh0{z) = — - ^ ( l ; / ? ; * )
and T{0)
Eap{z) m + zEa,a+l3(z)
(1.20.7)
where 1.F1 is the confluent hypergeometric function defined by (1.17.4).
The integral representation of Ea^(z) is given by
_ 1 r t°-0ez
En 3 = -—: / dt , P 9.7T7 JC tP - 7.
(1.20.8) 27T2 JC t° - Z
where c is the same path as in (1.20.3). Similarly the Laplace transform
of t^~lEa(t°) can be obtained by means of integral
|o°CVV3"1 EQ{taz)dt = —^ (1.20.9)
where the region of convergence of (1.20.9) is the same as that of (1.20.4).
The functions Ea(z) and EQ^(z) increase indefinitely as z —> oo in a
certain sector of angle an, and approach zero as z —>• oo outside of this
sector.
A function intimately connected with Ea0 is the entire function
4>{CL,Q,Z) = £ iS, k\ T{ak + (3)
, Q , / ? > 0 (1.20.10)
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Applied Mathematics A. M. U. Aligarh
which was used by Wright in the asymptotic theory of partitions. The
asymptotic behaviour of (p(z) as z -> oo was also investigated by Wright
Here one can easily verify that
Uz) = ( l y ^ l . i z + l; - j j (1.20.11)
It shows that Wright's function may be regarded as a kind of gener
alized Bessel function Jv{z), defined by equation (1.20.2).
(f>(z) can be represented by the integral
(f){a,P]z) = —I u~p exp(u + zu~a) du, a > 0 (1.20.12) Z7TI J-oo
The methods developed here are shown to apply not only to Laguerre
polynomials and hypergeometric functions but also to such other special
as Mittag-Leffler's function E0, Ea^ and Wright's function (j)(a,P; z).
1.21. GENERATING FUNCTIONS
A generating function may be used to define a set of functions, to
determine a differential recurrence relation or pure recurrence relation to
evaluate certain integrals etc. We define a generating function for a set
of function {fn(x)} •
Definition: Let G(x, t) be a function that can be expended in powers of
t such that
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Applied Mathematics A. M. U. Aligarh
oo
C?(x,t) - Y,Cnfn(x)tn (1.21.1) n=0
where cn is a function of n, independent of x and t. Then G(x, t) is called
a generating function of the set {/n(^)}- If the set of function {fn(x)} is
also
defined for negative integers n = 0, ±1,±2, •••, the definition (1.21.1)
may be extended in terms of the Laurent series expansion
OO
G(x,t) = £ Cnfn(x)tn (1.21.2) n=—oo
where {c^} is independent of a; and t. The above definition of generat
ing function used earlier by Rainville [122, p. 129] and McBride may be
extended to include generating functions of several variables.
Definition: Let G(x\ • • -Xk,t) be a function of (k + 1) variable, which
has a formal expansion in powers of t such that
oo
G{xl,---,xk„t) - '£cnfn(xi,---,xk)tn (1.21.3)
where the sequence {cn} is independent of the variable Xi, • • • ,Xk and t.
Then we shall say that G(x\, • • •, Xk, t) is multivariable generating func
tion for the set {/„(xi, • • •, Xfc)}JILo corresponding to non-zero coefficient
{cn}-
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Applied Mathematics A. M. U. Aligarh
Bilinear and Bilateral Generating Functions
A multivariable generating function G(x\ • • • xk, t) given by (1.21.3) is
said to be multilateral generating function if
fn{Xi---Xk) = gi0l(n){Xl) • • • 9kak(n){xk) (1.21.4)
where ocj(n), j = 1,2 • • •, k are functions of n which are not necessarily
equal. Moreover, if the functions occurring on the right hand side of
(1.21.4) are equal the equation (1.21.3) are called multilinear generating
function.
In particular if
00
G{x,y;t) = E cnfn{x)gn{y)tn (1.21.5) n=0
and the sets {fn{x)}™=o an<^ {#n(?/)K£Lo a r e different the function G(x. y\ t)
is called bilateral generating function for the sets {/n(^)}^=o o r {9n{y)}^Lo-
If {fn{x)}%Lo and {gn{y)}^Lo are same set of functions then in that
case we say that G(x, y; t) is bilinear generating function for the set
{/«W)nM=0 0 r {0n(y)KLO-
Example of Bilinear Generating Function
(l-O-'-exp = M ofl _; 1 + a; (i-ty
0 0 = n\ L*(x) L°n(y)f
nto (l + a)„ (1.21.6)
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Applied Mathematics A. M. U. Aligarh
Example of Bilateral Generating Function
(l-t)-l-c-° (l-t+yrc) exp [0^ ^
OO
= E 2*1 [-n, c; 1 + a; y] Z#>(x) tn (1.21.7)
1.22.INTEGRAL TRANSFORMS :
Integral transforms play an important role in various fields of physics.
The method of solution of problems arising in physics lie at the heart of
the use of integral transform.
Let f(t) be a real or complex valued function of real variable t, de
fined on interval a < t < 6, which belongs to a certain specified class of
functions and let F(p, t) be a definite function of p and £, where p is a
complex quantity, whose domain is prescribed, then the integral equation
0[/WiP] = J*F(p,t)f(t)dt (1.22.1)
where the class of functions to which f(t) belongs and the domain of p
are so prescribed that the integral on the right exists.
F(p, t) is called the kernel of the transform (j>[f(t),p], if we can define
an integral equation
f(t) - JdF(t)<t>[f(t),p]dp (1.22.2)
then (1.22.2) defines the inverse transform for (1.22.1). By given different
values to the function F(p, t), different integral transforms are defined by
61
c; 1 -I- a; xyt
(l-t)(l-t + yt)
Applied Mathematics A. M. U. Aligarh
various authors like Fourier, Laplace, Hankel and Mellin transforms et
cetera.
Fourier Transform
We call
•H/M; fl = (270~1/2 rjWe*x dx (L22-3) the Fourier transform of f(x) and regard x as complex variable.
Laplace transform
We call
/*0O
C[f(t)\ P) = /0 f (*)*'* d* (1-22.4)
the Laplace transform of f(t) and regard p as complex variable.
Hankel transform
We call
7C[/W; 4] = /0 / W a ( ^ ) d* (1.22.5)
the Hankel transform of f(t) and regard £ as complex variable.
Mellin transform
We call
M[f(x); s] = I f(x) xs~l dx (1.22.6)
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Applied Mathematics A. M. U. Aligarh
the Mellin transform of f(x) and regard s as complex variable.
The most complete set of integral transforms are given in Erdelyi et
al. Ditkin and Prudnikov and Prudnikov et al.
Other integral transforms have been developed for various purposes
and they have limited use in our work so their properties and application
are not mentioned in detail here.
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Applied Mathematics A. M. U. Aligarh
CHAPTER II
ON SOME FINITE INTEGRALS
INVOLVING GENERALIZED
POLYNOMIAL SETS AND THE
MULTIVARIABLE i*-FUNCTION WITH
APPLICATIONS ABSTRACT: The present chapter deals with the evaluation of the three
new finite integrals involving the product of generalized polynomial sets
and the multivariable //-function. These integrals are unified in nature
and as a key formulae from which one can derives as its particular cases,
integrals involving a large number of simpler special functions and poly
nomials. For the sake of illustration, only one particular case of the
integrals obtained has been given which is also new and is of interest.
At the end, application of the main results have been given by inter
connecting them with the Riemann-Liouville type of fractional integral-
operator. The results obtained are basic in nature and are likely to find
useful applications in several fields notably electrical network, probability
theory and statistical mechanics.
Applied Mathematics A. M. U. Aligarh
2.1. I N T R O D U C T I O N :
The multivariable //-function of several complex variables Srivastava
and Panda ( [155]. see also [160]) is represented by means of the multiple
Mellin-Barnes integrals in the following form :
H[zi,---,zr] TTO,N:m.ini;-;mr,nT n P,Q:pi,q\\-,Pr,qr
Z\
L *T
Sr) .(»•) Irh (a,; a), • • •, ay) : (c), 7J)i,Pl; • • •; (c)r\ 7}r))i,Pl
= ^yL-LM^-^^
X ^ ( 6 ^ - - ^ r ) 2 ? r , - - - , 4 r ^ l - - ' ^ r , (2.1.1)
where UJ = \/—T
M&) =
ranr(4"-^)"rir(i-cf + 7j1)«i)
9i ft., r(i - df] + 6fki) _ri r(<f - 7i&) /=mi + l
(2.1.2)
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Applied Mathematics A. M. U. Aligarh
V e { l , - , r }
< />&• • •& • ) =
fi r(i -aj+t af^i) 3=1 j=l
9i „ , r n i ^ . . Pi We.\ ft TYI _ h. _L f /?(!), n r(a,- - E a '&) n r(i - 6,- + E &1}& (2.1.3)
To develope the //-function of /c complex variables z\, • • • ,z2- All the
Greek letters are assumed to be positive real numbers for standardization
purposes; the definition of the multivariable if-function will however, be
meaningful even if some of these quantities are zero.
For the convergence of the integral given by (2.1.1) and other details of
the multivariable //-function one is referred to the book by Srivastava et,
al. ( [64], pp.251-253, eq. (c-l),(c-5) ).
The generalized polynomials (multivariable) defined by Srivastava [163]
in the following manner:
o o lQl/Pl] [QR/PRUQI)P (OD)O GPI,-,PR r~ ~ 1 _ V . . V v W f l a i \WR)PRaR °Oi , - ,Qn *' ' ' >xr\— 2^, 2 ^ ~~j j
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Applied Mathematics A. M. U. Aligarh
xA[Q1,a1,---,QRaR] x?1, • • - , < ' , (2.1.4)
where Qj, Pj ( j = 1, • • • > R ) are non - zero arbitrary positive inte
gers. The coefficients A[Qia.\, • • •, QROCR] being arbitrary constant, real
or complex.
If one takes R — 1 in the equation (2.1.4) and denote A [ Qi, a j thus
obtained by Aji,an o n e arrives at the well known general class of poly
nomials SQ [X] introduced by Srivastava [147].
By suitable specializing the coefficient AQ0 the general class of polyno
mials can be reduced to classical orthogonal polynomials, Bessel polyno
mials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials,
Gagenbauer polynomials and generalized hypergeometric polynomials.
The generalized polynomial set defined by Raizada [123, p.64, eq.(2.1.2)]
in the following Rodrigues type formula:
S%Ar [X, r, s, q, A, B, fc, 1} = S°AT [X] = {Ax + B)-°( l - rx)r~^T
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Applied Mathematics A. M. U. Aligarh
x7T)+n [{Ax + B)a+qn (1 - Txr)(-0/T)+m] (2.1.5)
with the differential operator Tk,i is defined by
Tk,i = xl{k + x—). (2.1.6) ax
Though the polynomial set S%,(3,T [x] can also be derived from the general
sequence of function defined by Agarwal and Chaubey [3] (see also [162],
problem 16,p.447), it unifies and extends a number of classical polyno
mials studied by various research workers such as Krall and Prink [95],
Gould and Hooper [51], Singh and Srivastava [131],Chatterjea [29] etc.
Moreover it can be expressed in the following series form as given in [123,
p.71, eq.(2.3.4) ].
S%AT [X, r, s, q, A, B, m, k, 1} = Bqnxl{m+n){l - rxr)sn l{m+n)
m+n f m+n f {-l)S(-p)t{-S)i(a)s{cL-qn)i h h h h p\SM\t\(l-a-6)i
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Applied Mathematics A. M. U. Aligarh
(-(3 \ (i + k + rt\ (-Txr\p(Ax\6 ,ni , x I" - s n ) P (——U\T^P) {•¥) ' (2L7)
Taking A = 1,5 = 0 and r -» 0 , one arrives at the following polynomial
set:
m+n p SZ'0'0 [x, r, q, 1,0, m, fc, /] = ^ n + ' ( m + n ) /(rn+n> £ £
p=0 «=0
^ ( • i * ^ ) ( , L 8 )
2.2. T H E MAIN INTEGRALS :
In this section the following integrals have been derived:
T H E F I R S T INTEGRAL :
£{x - a)p-\b - x)a-\x - c)-p-aS^° (x — a y[
\x — c
b — x . ;r, q, 1,0, m,k,l x — c
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Applied Mathematics A. M. U. Aligarh
qPi,-,Pn x — a\ (b — x \x — c \x — cj \x — c \x — c
x — a\ (b — x
xH \x — cj \x — cj \x — cj \x — c
dx
— (b — a)P+a+(t1+l/)\<ln+l(rn+n)+rP}~1 (l) _ c}-/9-/'[gn+<(«x+n)+rp]
x(a — c ) _ o r - 1 / l 9 n + K m + n ) + r P ] rm+n qn+l{m+n)+rp
m+n * T , A-p)flP (a + qn + k + rp
p=0 / = 0 P- J • \ l ' m+n
TJO, N+2 •.m.in\;--;Tnr,nr nP+2,Q+2 :pi,qi;-\pr,qr *iter(fctr.-.*(fer(^r
„(rh. . (A «,i Fi, F2; (a,; a}, • • •,ajOi,? : (cj,7J)i,Pl; • • •; {c)\fp)i^
?1 tf(r)\ „ . (A\ Xl (r) X{r)> (6,-,/?), • • • ,/?}")i,Q : (d},J})iA; • • •; ( # ' . * ; )i*5 ^3
, (2.2.1)
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Applied Mathematics A. M. U. Aligarh
where
[Qu'Pi.
£(yi , - - - ,y*)= E [QR/PR] R
E n a 1 = 0 a r = 0 j'=l
1 Qj'JPjt^ji
Otji\
and
\ (b-a\ ' (b-a\ '" A[Ql,a1----;QRaR}. (2.2.2)
Fi = (1 — p — A*[9n + Km + n ) + rP] _ ^l^i
hflo:;? ; gi,---,gr) (2.2.3)
F2 = (1 — a — v[qn + l(m + n) 4- rp] — fciai —
/cfictij ; i y i , - - - , i o r ) ( 2 . 2 . 4 )
F3 = (1 - p - a - (fj, + v)[qn + l(m + n) + rp] - (/ij + ki)a\
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Applied Mathematics A. M. U. Aligarh
(hR + kR)otR ; gi + wi,---,gr + wr) (2.2.5)
provided that giWi > 0 V ie {1, • • •, r } , ; hj > 0, fcj > 0 V j ' e {1, • • •
• • •, R} Pj> (f = 1, • • •, R) is an arbitrary positive integer and the coeffi
cients A[Qi, cti; - • •; QR, ap] being arbitrary constant, real and complex.
Re > 0 , #e ^ + E ^ ' -rf>\
i= l *f/j > 0, Oi > 0,
where
| an?z |< - 7r Oj, c < a < 6,
fii = E 4' + E 7J0 - E T]" - E #' .7=71+1 j = l j=Tli + l J = l
^ Jf(») V - Jf(») + £ * } ' - E ^ > 0 V e ( l , . , r )
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Applied Mathematics A. M. U. Aligarh
THE SECOND INTEGRAL:
fQ{t - ^ f M p - IX)S£::PQRR [yi(t - *)fcl A
• • •, yR(t - x)h«xk"} SS^° [y(t - xfxv- r, q, 1,0, m, fc, /]
xff [21 (t - x)9lxw\ • • •, 2r(* - x)9rxWr] dx
ta+0+(fl+u){qX+i{m+X)+rp)+l £ £ > ( y i , . . . , 2 / f l ) r ( p + n + 1)
s=0
x(-n) s(p + (7 + n + 1)*
|m+A gA+/(ro+A)+rp m+A P {—p)f CP
X ( 7 t / 2 ) ' n!s! r(p + a + l) § ,? 0 " l U T "
^J+ <?A + k + rf\ 0 N+i :mln1;...ynr,nr x I / J , - " P + l . Q + l :pi,gi;-;Pr,9r
\ ' / m+A
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Applied Mathematics A. M. U. Aligarh
Zl tgi+m Dx\ (ay, a j , • • •, a(-\P : (cj, 7 j ) l i P l ; • • •; {cf\j\r))hPr
(by/?},•••,/?jr))i,(?: (d),<5])i,9l; • • •; (4r )> ^ T ) ) i * ; ^ 2 (2.2.6)
where
[Q1/P1] [Qa/Pn] R
D(yu---,yR)= E ••• E n a\=0 CIR=Q j ' = l
( gjQpy.oy <y 2(/ l /+^)ay
a^!
xA[Qi,ai;---;Qiiaii] (2.2.7)
Di = (—f3 — v(q\+l(m+\+rp)-kiai kRaR] W\,---,wr) (2.2.8)
D2 = {l-oc-(3-{fj,-\- i/){q\ + Z(m + A) + rp) - (hx + fci)ai
(hji + kR)aR ; p! + wi, • • •, gr + wr) (2.2.9)
and Pjp,a)(x)is the well known Jacobi polynomials. Then integral (2.2.6)
converges under the following (sufficient ) conditions:
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Applied Mathematics A. M. U. Aligarh
Re At) \6i /
> 0 , Re d^Y
>0,
ar9z l< « *" ii ^» > 0, 9i> 0, ^
V ze {!,-••, r},;hr, ky > 0 V je {1, • • •, fl}.
T H I R D INTEGRAL :
j£(* - x ) ° ^ ( l - ixl2)aPir\l - ix) 5g ; : .5 i [yi(t - s)*1 A
• • •, yfl(t - x ) ^ ] S}c'° [y(t - x ) ^ ; r, <?, 1,0, m, A;, /]
xH [zi(t - x)9lxw\- • •, yR(t - x)9rxWr] dx
tu r + A v f mf A (~P)/CP /<S + gA + fc + r / -
*=Op=0/=0 P]f- \ l ^rn+X
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Applied Mathematics A. M. U. Aligarh
(p + s + l)„(-<7 - n)8 f'ytV n! s!
O, N+l :m\ni\—;mr,nr v TV,, n, m y ' " + 1 :mini;-;m,
Zi t9l+m
z £flr+«V
Di; (aj-; a), • • •, a$r))i,p : {c}n})i^ ' ' ' 5 ( 4 ° ' T ^ ) * *
(6 i ; /?],••• , /3J r ,)i ,Q : (dj , <5j)i,9l; • • •; (4 r )> ^ r ) ) i * ; ^ J (2.2.10)
where
[Qi/Pi] [QR/PR] R
D(yi,--,VR)= £ ••• E n Ql=0 Qfi=0 j ' = l
( Qj'Jpjf,^ ity 2(^.,+fc;.)Qy
Qj/'. 2/y
x A [ Q i , a i ; - - - ; Q i i a i j ]
DT = (—/?-i/[<jfA+Z(ra+A)+rp]-fcia:i fc#a#; Wi,---,wr) (2.2.11)
D2 = {-l-a-0 (/z + i/)(gA + /(m + A) + rp) - (hi + fci)«i -
76
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Applied Mathematics N ^ ^ w ~ ~ ^ ^ A. M. U. Aligarh
- {hR + kR)aR ; gi + wu---,9r + wr) (2.2.12)
U = a + (3 + (fi + u)[qX + /(m + A) + rp] + 1 (2.2.13)
and
V = pA + /(m + A) + rp. (2.2.14)
The integral (2.2.10) converges provided the condition mentioned with
the integral (2.2.6) are satisfied and the series occuring on the right of
(2.2.10) is absolutely convergent.
P R O O F :
To establish the integral (2.2.1), express the generalized polynomials
occuring on the L.H.S. in the series form given by (2.1.4), the generalized
polynomial set by equation (2.1.8) and the multivariable H-function in
volved there in terms of Mellin-Barnes contour integral by (2.1.1). Now
interchange the order of summation and integration (which is permissible
under the conditions stated),so that the L.H.S.of (2.2.1) ( say A) assume
the following from after a little simplification:
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Applied Mathematics A. M. U. Aligarh
\QilPy [QR/PR] R
E n «1=0 Q/J=0 j ' = l
( Qj'/py&y ny
C*j'! S/l j4[Qi,a:i;---;Q/i<*fl]
x/ rn+n # ^ / ( m + n j + r p " f " * (-p)fPP (<* + qn + k + rf^
m+T?
x I f (x — aY+(ii<in+l(Tn+n)+rP[+'Ef=ohJa]+T>rt=o9x^-1
x(b — xY+u^qn+l^m+n^+rp^+^=ok3C':'+^=oWl^~l
x(x — c)_p_a(M+^[9n+^m+n)+rpl+^f=i^+fc^Qj_^'=i(ff ,+w,^ idx} d£. (2.2.15)
On evaluating the inner integral occuring on the R.H.S. of (2.2.15) with
the help of known result [ 32, p. 210 , eq.(2.2) ] or [ 49, p.287, eq (3.119)],
one gets after a little simplification:
= (b — a)p+e7+^+^qn+l(m+n)+rp\~l(b — c)-p-^[<in+l(m+n)+rp\
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Applied Mathematics A. M. U. Aligarh
x ( _ \-<T-v[qn+l(m+n)+rp]rm+n qn+l(m+n)+rp
m+? P {~p)fl3p fa + qn+k + rf" X E E , r, I i
p=o /=o P- J- ^ L m+n
[Qi/Pi]
x £ a i=0
[Qfi/Pul R
E n a f i=0 j ' = l
(~9j')p/.a./ [ fb- a\n>' fb- a\"y
ar\ ,6 — c/ \6 — c,
xA[Qi,ai;---;Qiia/z]
(2TTU;) ru>)r ^1 ^ r T[p 4- v{qn + l(m + n) + rp) + /ii^i + • •
T[p + a + (// + J'Xq'n + l(m + n) + rp)
x + hRaR]gi£i + ••• + grZr] r[(<7 + i/(gn + /(m + n) + rp)
+(/ii + fci)ai + + (hR + kR)aR] (gx + Wi)$i +
+kxai H 1- fcflQ;fi; w ^ H h wr£r)
*(t>l{Zl)---<i>r{Zr)^l, •••,&) Z\ b-a\91 fb-a\Wl)Z
,6 — c) \b — c
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Applied Mathematics A. M. U. Aligarh
•M ns dCi, •••<£.. (2.2.16)
Finally, on reinterpreting the Mellin-Bannes contour integral in the R.H.S.
of (2.2.16) in term of the multivariable H-function given by (2.1.1), one
easily arrives at the desired result (2.2.1).
To derive integral (2.2.6), express the generalized polynomials by
equation (2.1.4) and the generalized polynomial set by equation (2.1.8)
and write the multivariable H-function occuring in the integral and in
term of Mellin -Bannes contour integral with the help of(2.1.1) and inter
change the order of summation and integration, which is justified as the
series involved is finite.
Then apply a known result [37,p.l92,eq(46)] in the order to evaluate the
x-integral and interprete the resulting contour integral as multivariable
H-function with the help of the (2.1.1), the integral formula follows at
once .
The third integral (2.2.10) can be proved in a similar manner as indicated
above. Here use in made use of the result [37,p.l92, eu.(48)]in order to
evaluate the rr-integral and to the resulting contour integral is interpreted
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Applied Mathematics A. M. U. Aligarh
as multivariable //-function to arrive at the desired result.
2.3. SPECIAL CASES :
For N = 0 = F, Q = 0, the multivariable H-function breaks up into
a product of r //-function and consequently, (2.2.1), (2.2.6) and (2.2.10)
reduce to
f\x - a)P-\b - x)a-\x - c)-p-° Ja
£0,0,0 (x-ay (b-x\v
y\ ; r , g , l , 0 , m , M \x — c) \x — c)
v *PI,-,PR y\
x — a\hl fb — xs x
x — c \x — c> ,••• ,VR
-a\hri /b-x\kR
x — c) \x — c
o-m„n, p,&
'x — a\9, fb — x wl
,(0 „,(0
,x — c \x — c,
\^J ' 'J A-P.
(<f.f)u. dx
= (6 — a ) / ' + a + ( ' x + , ' ) l 9 n + / ( m + n ) + r P ] - 1 ( 5 _ c-\-p-fi\qn+l(m+n)+rp}
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Applied Mathematics A. M. U. Aligarh
/ _ \-(T-u[qn+l(m+n)+rp]jm+n qn+l(m+n)+rp
™+? P _, d~P)f Pp fa + qn + k + rf" x 5 s^'"-^-^ P\f\ i m+n
rrO,l :mi,ni-1-;mr,nr A / 7 2 , l :pi,gi;-;Pr,9r £l \x-c) \x-c) ' " " ' ' Zr \x-c) \x-c)
Fi,F2;(c},7J)i,pi;---;(4r),7f)i,pr
(d),8})l,qi;---;(d{;\6\%^F3
(2.3.1)
The condition of validity of (2.3.1) can be easily obtained from those of
(3.2.1). L(yi, • • • ,yn), Fi, F2 and F3 are given by (2.2.1),
(2.2.3), (2.2.4) and (2.2.5) respectively.
£(t-x)"x*P<r\l-ix)
52:---:8i M* - ^ A • • • • **(* - a)*****]
82
. ^ i k i J3*
Applied Mathematics A. M. U. Ahqarh
SSx^{y(t-xyxv;r,q,l,0,m,k,l}
> < n { frmt,nt
Pi,Qt zzt
9' 1+u. dx
,3=0
yf ,s j^m+X yqX+l(m+X)+rp
n!s!r((p + s + l)
p=C /=0 P] P- ^ l 'm+A
rjO,l .rrn,ni;- ;rar,nr x - n i , i -pi^i;-;Pr,9r
2 l t s i + « i
zr t9r+Ur
Dvic1 -y1^ •••••(rc(r) 4r)\,
(2.3.2)
The condition of the validity of (2.3.2) can be easily obtained from those
of (2.2.6), D(yu---,yR), Di and D2 are given by (2.2.7), (2.2.8) and
(2.2.9) respectively.
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Applied Mathematics A. M. U. Aligarh
ft(t-xrj(i-^)°p}r\i->rx)
xSk'-'Z [yi(* - *)*• A • • •. y*(* - ^ ' ^ J
xSt'c '°[y(t-x)^;r,g,l,O fm,fe,Z]
i = l
TTmt>nt *i(* - a;) V ' (C{ i) 7 ( i ) ) 1
dx
s=o P=o /=o P-f- \ I i m+X
(p + s + l)„(-g - n). fyf
rrO,l : m i , n i ; - ; m r , n r X / 1 1 , 1 :pi,qi;-;Pr,qr
Zi t » 1 + W l
^ ^Sr+Wr
A;(c),7J)i,P l;---;(4 r ) ,7l r ))i,P ,
(2.3.3)
The condition of validity of (2.3.3) can be easily obtained from those of
(2.2.10), D(yu • • •, yR), Du D2, U and V are given by (2.2.7), (2.2.11),
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Applied Mathematics A. M. U. Aligarh
(2.2.12), (2.2.13) and (2.2.14) respectively.
For r = 1, N = 0 = P, Q = 0 the multivariable //-function reduces
to Fox's //-function and one gets finite integrals involving generalized
polynomial, general polynomial set and Fox's //-function.
If one takes R = l in the main integral, one gets the finite integral involv
ing a general class of polynomials (studied by Srivastava [147]), general
polynomial set and the multivariable //-function.
Also suitably specializing the parameters of the multivariable H-function
and the generalized polynomial sets in the main integrals one can further
obtain a large number of new and known integrals as their special cases.
2.4.APPLICATION:
The Riemann-Liouville derivative of function f(x) of order a (or alterna
tive ath order fractional integral) [37,p.l81;llp.49] is defined by
aD?{f(x)}={ (2.4.1) dq
aDff
x-q{f(x)}, q-KRe(a)<q, I dxi
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Applied Mathematics A. M. U. Aligarh
where q is positive integer and the integral exists.
For simplicity the special the special cases of fractional derivative operator
aD~a when a=0 will be written as Dax. Thus
D-° = oD: (2.4.2)
Now by setting b=x in the integral (2.2.1), it can be rewritten as the
following fractional integral formula;
aD°x{{t-ay-\t-c)-o-°
^^^(^"(H)^'^1'0'771'^^
XbQw;Qn t-a\hl /x-t\k
*\—c) \—c) ' " ' ^ U - c , vt-c . t-a\hR (x-t\kli
H z\ t-a\9x /x-t\Wl
t — cj \x — c , zr
't-a\^ (x-V 1//VT
X — CJ \X — CJ dx
I» (x — a ) p + C T + ^ + I / ^ 9 n + ' ( m + n ) + r ^ ( x — c)-p-v[qn+Km+n)+rp}
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Applied Mathematics A. M. U. Aligarh
X(a — r)~(T~l'{qn+l(Tn+n}+rp\ / m + " Qqn+l(m+n)+rp
m+n £ r , A-p)fpP (a + qn + k+rr E £ L{yi, •••,yR) — - j — -
<p=0 / = 0 P-J- \ l /m+n
TTO, iV+2 :mi,ni;—;mr,nr nP+2,Q+2 :pi,9i;-;Pr,9r
(x^\91 (x=a\Wl y (ZZ<L\9T fx-aY"r
*l\x-c) \x-c) ' ' " ' ^ V i - J \x-c)
» FuF2](aj-a}---a(p)hP:(c)n})Upi].-.A4\j}%
(r) X{rh (bj, (3\ • • • $ \ Q : (d), 6})W, • • •; (df\ 6f\qr; Fs
(2.4.3)
where Re(a) > 0 and all the conditions of validity mentioned with (2.2.1)
are satisfied. L(yi, • • • ,yR), F l5 F2, and F3, are given by (2.2.2), (2.2.3).
(2.2.4) and (2.2.5) respectively.
By setting t = x in (2.2.6). the integral can be rewritten as the following
fractional integral formula:
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Applied Mathematics A. M. U. Aligarh
D-a+l ( ^ ) ( 1 _ J t ) sAr-vF* [yi(a. _ ^ ^
y*(z - t)h*1**} Sf c'° [y(x - t)» tv• r, q, 1,0, ro, k, 1}
xH [zi(x - t)gHw\- • •, zr{x - t)Hkr] dx
_L_ ( x ) a + /3 + (^ ) [ 9 A + / ( r a + A) + r p] £ D{yu ...yR)
s=0
xT{p + n + l)(-n)s(p + <r + n + l) s f ^ -y£\ s £ra+A yq\+l(m+\)+rp
n\s\T{p + s + 1)
"£A £ ( -P) / Cp /^ + gA + fc + r/^ ~ *+1 mi,„1,.,Wpfl, P=o /=o P ! / ! V /
H m-f A
P+1,Q+1 :pi,9i;--,Pr.9r
Zi tf1+Wl
z igr+wr
A ; fa; a) • • •, a J V = (<$• 7])i,Pl; • • •; (cf,7jr))i,pr
fe01, • • •,0jr))i,o : (<*},<5j)i,9l; • • •; (d$r), tfV; D, (2.4.4)
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Applied Mathematics A. M. U. Aligarh
where Re(a + 1) > 0 and all the conditions of validity mentioned with
(2.2.6) are satisfied. D(yu- • • ,yR), Dl and D2, are given by (2.2.7),
(2.2.8), and (3.2.9) respectively.
By setting t = x in (2.2.10). the integral can be written as the following
fractional integral formula:
D-^^[l-^)ap^\l-lt)
XSQZ;QH HX -1)*1**1. • • • > VR(* - t)hRtkR]
Mfi xSo^[y(x-tre;r,q,l,0,m,k,l}
xH[Zl{x - t)Hv\ • • •. zr(x - t)HWr] dx
.u im+X yV f " A f i-P)f(P (S + g\ + k + rf\
' r(a + i)y h^h p}-f]- v i J,
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Applied Mathematics A. M. U. Aligarh
( l x \ n / \TTO,N+1 :mun1;--,mr,n (p + S + l)n{-<J-n)s(7Xy ^ .._\jjOtN+l :mx,n1-,..-mT,nT
n! s\
Zi t9l+Wl Di; {ay, a), • • •, aj-r))i,P : (cj, 7J)i,Pl; • • •; (c$r), 7Jr))i,Pr
(&,•; /?\ • • •, $ \ Q : (dj, 5))^; • • •; (df\ #>)1|fr; D2
(2.4.5)
where Re (a + 1) > 0 and all the conditions of validity mentioned with
(2.2.10) are satisfied. D(yl, ••• ,VR),DI D2, U and V are given by (2.2.7),
(2.2.11), (2.2.12), (2.2.13) and (2.2.14) respectively.
The fractional integral formula given by (2.4.3), (2.4.4) and (2.4.5) are
also quite general in nature and can easily yield Riemann-Liouville frac
tional of a large number of simpler functions and polynomials merely
by specializing the parameters of multivariate H-function SQY'.'.!^ and
ga,/3,o o c c u r m g m it which may find applications in the theory of proba
bility.
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Applied Mathematics A. M. U. Aligarh
CHAPTER III
iV-FRACTIONAL CALCULUS OF
GENERALIZED iJ-FUNCTION
ABSTRACT : The present chapter deals with Nishimoto's iV-fractional
differintegral of the I-function whose argument involves the product of
two power functions (z — a)~x and (z — b)~^ (A, ft > 0).On account of the
general nature of the main result, iV-fractional differintegral of a large
variety of special functions having general argument follows as special
cases of the present findings. Results of Romero, Pricto and Nishimoto
[126], Romero, Kalla and Nishimoto [125] and Gupta, Goyal and Garg
[52] follow as special cases of the present study.
3.1. INTRODUCTION :
The iV-fractional differintegral a function given by Nishimoto [114] is
defined as follow; Let D ={D-,D+},C = { C _ , C + } ,
C_ be a curve along the cut joining two points z and — oo +a; Im(z),
Applied Mathematics A. M. U. Aligarh
C+ be a curve along the cut joining two points z and oo + u Im(z),
D- be a domain surrounded by C_, D+ be a domain surrounded by C+.
(Here D contains the point over the curve C).
Moreover, let / = f(z) be a regular function in D, then the fractional
differintegral of arbitrary order v € R (derivatives of order v for v > 0
and integrals of order v for v < 0) of the function f(z), if | (/)„ | exists,
is defined by
F„ = (/), = ,(/)„ = r±±21 lJ-M-ld< („ t z - ) (3.1.1)
where — n < arg(( — z) < n for C_,0 < arg(( — z) < 2ir for C+,C ^ z
,and T is The Gamma function.
The I-function introduced and studied by Saxena occuring in this
chapter is defined as follows[129]:
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Applied Mathematics A. M. U. Aligarh
'W = VIM = Z&r ^(2j, OCjjltn) (Oji, C^jijn+^p,
(Oj, Pj)l,rni {Oji, fJji)m+l,q,
= i/,*^ Here and in what follows UJ stands for v ^ l , and
(3.1.2)
m n r(6,- - ft-o n r(i - fli + a,-o xg) = - J^ ^ -
E{ ft {rfi-fy + z^)} ft rCoji-ajiO} i = l j = m + l j = n + l
(3.1.3)
Pi(i = 1, • • •, r), qi(i = 1, • • •, r), m, n are integers satisfying 0 < n <
Pi, 0 < m < qi,(i = l , - - - , r ) , r is Hmte,aj,0j,aji,Pji are real and
positive and aj,bj,aji,bji are complex numbers such that
atj(bh + v) ^ 0h(a>j ~ 1 -k);h = l , - - - , ra ; j = l , --- ,n.
The integral on the right hand side of (3.1.2) is absolutely convergent
when Q{ > 0 and | argz |< —7rr2z, where Li
m n qt p.
Vi = E f t + E « i - E ft-*- E Oi*>o. (3.1.4) j = l j = l j = m + l j = n + l
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Applied Mathematics A. M. U. Aligarh
3.2. MAIN RESULT :
((* - aY(z - bfI\A{z - a)~\z - b)%
00 fa — b\k 1
= e-~(,_tr~£g_)_
Tm,n+2 Xip,+2,9i+2,;r A(z - b) - />V'~A
(l-v + p + a-k-,\-p),
\Pji Pjjl.mi \OjiiPji)m+l.qti
(1 + p - k, A), (aj, aj)i,n(°jti ^)n+i,P.
(1 + p + a - k, A - /x), (1 + p, A) (3.2.1
provided that the I-function occuring on the right hand side of equa-
tion(3.2.1) exists,
a — b z 7 a, 6 and
z-b I<1
Re (p + a) - (a - p) max Re [— 1 < p. < 0 l<J<n a,
and
(a - /i) > 0. (3.3.2)
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Applied Mathematics A. M. U. Aligarh
P R O O F :
First applying the definition of fractional differintegral given by (3.1.1)
in the left hand side of (3.2.1), then expressing the I-function in terms
of contour integrals given by (3.1.2) and interchanging the order ( and £
integrals one collects the terms involving the powers of (z — a) and (z — b)
and writes them in term of the power of (z — b) as follows:
{z _ ar*[z - t r * = £ ( -p-y- t ) ' fc=0 fc!
x {z - &)P+"+(M-AK-k (3.2.3)
provided
Zy^a.b and I r 1< 1. z — b
Finally one evaluates £ integral, which is iV-fractional differintegral of
order v of power function (z — &)p+<T+^-A^-fc.
Then following the method given by Nishimoto [113,vol.5,p.l,eq.(l)] one
obtains the left hand side of equation (3.2.1) as follow:
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Applied Mathematics A. M. U. Aligarh
j-/*o*£<2^a^.-~ fc=0
valid under the conditions
Re(p + <r)-(a-n) max R e [ ^ — ] < /x < 0, (A - /i) > 0. (3.2.5)
Now interpreting the result thus obtained in terms of I-function, one easily
arrives at right hand side of (3.2.1).
3.3. SPECIAL CASES :
(I) If one takes r = 1 in equation (3.2.1), the following Ar- fractional
differintgral of Fox's #-function [105,160] is obtained.
((z - aY(z - bYH[A(z - a)-\z - b)%
00 /
e-u*"(z - b)p+a-v £ fc=n V
a-b\k 1
ktQ\z~bJ k\
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Applied Mathematics A. M. U. Aligarh
TTm,n+2 ""pi+2,<Zi+2,;l A(z - by~x
(1 - v + p + u - k, A - p),
(fy> ft)i^n (1 + P + <7 - M - AO,
(3.3.1) (l + p-fc,A),(aj,«j)i,pi
(1 + p + a - k, A - p), (1 + p, A)
The result holds true under the condition easily derivable from those
stated with the main result.
(II) If in equation (3.2.1), one takes a = 6, a — 0,p = 0, the following
result is obtained after a little simplification.
((* - aYI[A{z - a)-x\)v = e-™(z-a)P-v
i-m,n+l 1pl+l,ql + l,;r A{z~a) -X
(1 - v + p, A), {a,j, Qj)i,„(aji, aji)n+hPi
{bj,Pj)l.m, {bji, Pji)m+l.qi, (1 + P, A) (3.3.2)
(III) If one takes r = 1 in the above equation (3.3.2) and use the re
sult [160,p.l4 eq.(2.2.16)] one gets the following result after a little
simplification:
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Applied Mathematics A. M. U. Aligarh
{{z-aYH[A{z-a)x\)v = e~"™(z - a)<-v
Tjm+l,n A(z - a)' (aj,a ;)i,Pl,(-p,A)
(3.3.3)
(IV) On taking a = 0 in the above result equation (3.3.3), one get the
known result recently given by Gupta et.al. [52,p. 10, eq. (4.3)].
(V) On suitably specializing the parameters of .//-function occuring in
equation (3.3.3) to different Meijer's (^-function, one can easily derive
five known results given by Romero et.al. [125,p.36,37,eq.(10),(ll),(12),
(13)and(14)].
(VI) If one takes p = 0 = a in equation(3.3.3) and reduce the Fox's H-
function to generalized hypergeometric function [160,p.l8, eq.(2.6.3)],
one arrives at another known result given by Romero et. al. [126,p.
57, eq. (5)] after a little simplification.
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(VII) On suitably specializing the parameters of the //-function occuring
in equation (3.3.3), one gets the corresponding fractional differinter-
gral of G and R function introduced by Lorenzo and Hartlay
[99] and expressed in terms of //-function Gupta et. al. [52,p. 12,
eq. (4.11),(4.12)].These results may find application in varies fields
of science and engineering [100,101].
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Applied Mathematics A. M. U. Aligarh
CHAPTER IV
FOURIER SERIES INVOLVING THE
ff-FUNCTION
Abstract : The present chapter deals with an application of an integral
involving sine function,exponential function, the product of Kampe de
Feriet function and the //"-function to evaluate three Fourier series. A mul
tiple integral involving the //"-function has also been evaluated to make
its application to derive a multiple exponential Fourier series. Finally,
some particular cases have been discussed.
4.1. INTRODUCTION:
The //-function occuring in this chapter will be defined and repre
sented in the following manner [18]:
{ah otji Aj)hN, (a,, aj)N+hP
(Pj, /?J)I,M, (bj, fa Bj)M+1>Q
f,Arr Hp£[z] = H
M,N P,Q
Applied Mathematics A. M. U. Aligarh
where
M N
nnbj-wnmi-aj + ait)}** m = -jr — p (i-1-2)
n {r(i - 6, + ftfly%. n r(o, - a ^ ) j=M+l j=N+l
which contains fractional powers of some of the gamma functions.
Here, and throughout the chapter a,j(j — 1, • • •, P) and bj(j = 1, • • •, Q)
are complex parameters, ctj > 0 (j = 1, • • •, P) , fy > 0 (j = 1, • • •, Q)
(not all zero simultaneously) and the exponents Aj (j = 1, • • •, JV) and
Bj (j = M + 1, • • •, Q) can take on non-integer values.
Buschman and Srivastava [18] has proved that the integral on the right
hand side of (4.1.1) is absolutely convergent when Q > 0 and | argz [<
-7rQ, where
M N Q P
& = YsPj + HAjaj- E PjBj- E «j>0. (4.1.3) j=l j=l j=M+l j=AT+l
For further details of //"-function one can refer the research paper of
Buschman and Srivastava [18].
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Kampe de Feriet hypergeometric function will be represented as follows.
p M Q
a
ai, • • • ,ap
bi,b'1i---,bAnb'\M
C\, . . . yCq
di,d'v--'idaid'tJ
xy
oo oo Jm+n _ v- y^ j=i j=i x y
" m=fln,0 _ ft (Cj)m+n _ft {(d,-)mK)»} m!n! (4.1.4)
j=M+l " j=l
(p + v <q + a+lorp + v = q + a + l and | x \ + \ y |< min(l,29~p+1));
1 r+ioo r+ioo
4ir2K /
+IOQ r+ioo .
_ioQ j_ioo ^tn-sn-tx-xyi-yyds dt, where
ft naj) ft {r orCftj)} K = - ^ 3— —
. n r{Cj) nAndjWj)} j=M+l j = l
(4.1.5)
and
fi (a, +s + t) ft {T{bj + s)r(6' + t)}
Ms, t) = - f 1 - 3—a n r(c3- + s +1) n {r^- + s)r(dj +1)} j = M + l j = l
(4.1.6)
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Applied Mathematics A. M. U. Aligarh
if put v = 0 = cr,then it changes in the following form;
F
P
q a
C\,. . . , Cg
\
xy
)
I
— F
\
au---i<h
C\, . . . , CQ ; x + y
(4.1.7)
For further detail one can refer the monography by Appell and Kampe
de Feriet[8].
Mishra [108] has evaluated
/ / (s in x) w—limx rp ap] C(sinx)2h
Pqi dx
•K^™'2 °° ( a p ) r C T ( « / + 2/ir) 2w~ i ^ - i (4.1.8)
rt0(pq)rr\4hrr("+2hr±M+1y where (a)p denotes au , av\ T(a ± b) represents T{a + b), T(a - 6);
h is a positive integer; p < q and Re(w) > 0. Recall the following
elementary integrals;
re*m-*xdx=r ' m=n;
•to 0 , m / n ; (4.1.9)
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f" Jo
jimx cos nx dx —
7T
2 7T
0
Jo jimx sin nxdx — <
m = n = 0 ; m = ii]
.7T i— , m = n ; 2
(4.1.10)
0 m ^ n ; (4.1.11)
Provided either both m and n are odd or both m and n are even integers.
Mishra employed (4.1.9), (4.1.10) and (4.1.11) to establish three Fourier
series for the product of generalized hypergeometric function and also
evaluated a double integral and double Fourier exponential series for Fox's
//-function.
The present chapter deals with an integral involving the product, of
the Kampe de Feriet function [8] and the //-function [18], and make
its application to derive three Fourier series involving the /f-function.A
multiple integral involving The .//-function has also been evaluated and
one derives a multiple exponential Fourier series involving the above H-
function.
Finally some particular cases have been discussed.
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For brevity, the following notations will be used.
n {ek)r+t n (A), ,n (fk)t fc=l fc=l fc=l _ f
G H H'
n (gk)r+t n (hk)r n (h'k)t fc=l fe=l fe=l
fci=i fci=i fci—l _
<?! Hi H[ ~ l
n ( p u j n + t . n (/lukjn n (^ifclk Kl = l fci = l fci = l
# 2 ^2 F 2 , n (e2A:2)r2+«2 PI (/2fc2)r2 II ( / ^ k fc2=l fc2-2 fc2=l _
G2 tf2 # 2 ~ 2
n (52fc2) n ( 2fc2)r2 n (ti2k2)t2 K2=i K2=i K2=i
II ( e „ f c n ) r n + t n II {Jnkjrn JIUnjJtn
~G ~H H~' rf {gnkjrn+tn U {hnkn)rn ,^(h'nkjtn
l*>Tl *• " ' f t *• " " f t "
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Applied Mathematics A. M. U. Aligarh
4.2. THE INTEGRAL
The integral to be evaluated are
f(sin*rv-i^: to; (/);(/'); « ( « « ^
xnM,N z(sin a;)2ff (Oj, Ctj; Aj)itN, {(Lj, Ctj)N+l,P
(bj, PJ)I,M, (bj, (3y, Bj)M+\,Q dx
yfln)eim^2 ™ (alV)r(PIV)1
2^-1 f-_e" r,t=0 Hi!
x / J P + l , Q + 2
(l-u-2pr-2>yt,2a;l),
(bj, PJ)I,M> (bj, f3j\ BJ)M+I,Q,
(a,j, ay, Aj)iyN, (aj, aj)jv+i,p (4.2.1)
provided that | argz j< -nQ, and Re(w) > 0; a, /?, /9,7, a, z are positive
integers, i = 1, • • • N 0 is denoted by (4.1.3).
Now making an application to (4.2.1),
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Applied Mathematics A. M. U. Aligarh
One derives a multiple integral.
f • • • f{sm x^-1 • • • {sin Xn)^-iei(m1x1+-+mnxn)
TpEi;Fi\F[ (ci); (/i); (/{); ' ai(sin xtf"
pEn;Fn;Fn Gn\Hn\H'n
(e„); (/„); (/n); an(sin xn)2pn
fen); ( M ; (^n); ^ ( « « £n)27n
X ^ M ^ [ (si11 ^i)2<Tl • • • (sin xn)2<Tn ] dxi • • • dzn
( 7 r ) ne
l ' ( r n i + - + m n ) 7 r / 2 °° °°
o ( w 1 + - + w n ) - n ^ * * • Zw ( e l * * " e n )
(01/4^(01/4*)** (OnlV-y-iPnlV*)** ri! ti! ' n- ^n-
X # M,N+n P+n,Q+2n 4(ai+- +a„)
(1 - wi - 2 p i n - 271*1,2<7i; 1)
(&j> ft)i,M, (fy, ft; Bj)M+hQ
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Applied Mathematics A. M. U. Aliyarh
(l-un- 2pnrn - 2jntn, 2an; 1), (a,-, ay, A,)i,jv, («J, Oj)iV+i,p
/ l-u)] -2pin -271*1 ± m : ^ | \ # / \-u)n-2pnr„-2intn±mn ^ . j \ (4.2.2)
provided that all the conditions of (4.2.1) are satisfied and Re(wi) > 0;
ai, Qi, &, pi, 7i, Zj are positive integers (i = 1, • • •, n)
P R O O F :
To prove (4.2.1) expand the //"-Function into the mellin-Barnes type inte
gral. Now on changing the order of integration,which is permissible under
the conditions stated with the integral, the integral readily follows from
(4.1.8).
On applying the same procedure as above the integral (4.2.2) can be
derived easily.
4.3. E XPON ENTIAL FOURIER SERIES :
Let
f{x) = (sin xr-lFg$%, (e); ( / ) ; ( / ' ) ; a(sinx)2<> (g);(h);(h')- (3(sinx)^
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Applied Mathematics A. M. U. Aligarh
frP,Q nM,N z(sin x) 2<T
(aj, otj\ Aj)i,N, (aj, OIJ)N+\,P
{bj> ft)i,Af. (bj, ft; Bj)M+I,Q
dx
0 0
= £ Ape-1^ (4.3.1) p=—oo
which is valid due to f(x) is continuous and of bounded variation with
interval (0,7r).
Now, multiplying by etmx both sides in (4.3.1) and integrating it with
respect to x from 0 to 7r, and then making an application to (4.1.9) and
(4.2.1), one gets
eim^2 ~ {a/wy {p/vy z r,t=0 rl t\
xH M,N+1 P+l .Q+1
z 4<r
( l - o ; - 2 p r - 2 7 t , 2 o - ; l ) ,
(bj, ft)i,M, (6j, ft; BJ)M+I,Q
(a.j, ay, A , ) ^ , (aj, aj)jv+i,p
(1-"-2^ r2-2^± m , (7;l)
. (4.3.2)
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Applied Mathematics A. M. U. Aligarh
An application to (4.3.1) and (4.3.2) gives the required exponential Fourier
series
u>—1 rpE;F;F' (2 sin xY-'FS^,
xnM,N z(sin x) 2(7 (a,-, Qj-; A j ) i ^ , (a j , OLJ)N+\,P
(fyi ft)i,A/, (fy> ft; BJ)M+I,Q
oo oo
p=-oor,<=0
^ / 2 - « ) , («/4T (P/vy '. e .
t\
frM,N+l xnP+l,Q+l
( l - w - 2 p r - 2 7 * , 2 ( 7 ; l ) ,
(fy, ft)i,M, (fy, ft; Bj)M+i,Q
(aj, Qj-; Aj)liiV, (a,-, ct^jv+i.p
( 1 - 0 , - 2 ^ - 2 7 ^ ^ . ^
(4.3.3)
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Applied Mathematics A. M. U. Aligarh
4.4. COSINE FOURIER SERIES
Let
/ (*) = (sin xr-lF% 1 jpE;F:F' G;H:H'
(e);(/);(/'); (<?);W; CO;
a(sin x)2p
(3(sin x)21
xH P,Q M,N
z(sin x) 2<7
(aj,OLJ\ Aj)ijf, (a,, aj)N+u
(bj,Pj)i,M, {bj, ft; Bj)M+i,Q .
* P=I (4.4.1)
Integrating both sides with respect to x from 0 to 7r, one gets
Bo i g c (ay (py 2 ^(Tr) rfco r! i!
x # M,N+1 P+l .Q+1
( ^ - pr - 7t, 2cr; 1), (a,, a,; A,-)I,JV,
(fy» ft)i,M,(fy, ft; BJ)M+I,Q
( a ^ Q j ) •j)«jVJV+l,P
(l^-2pr-270,(r;l)_ (4.4.2)
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Applied Mathematics A. M. U. Aligarh
Now, multiplying by elTnx both sides in (4.4.1) and integrating it with
respect to x frome 0 to 7r , and finally , making an application to (4.1.9),
(4.1.10) and (4.2.1), one derives
Br. ,ipw/2 oo
T E * (a/4T (0/V) y\t
* r,t=0 rl t\
xH M,N+1 P+l .Q+2
z 4"
( l - o ; - 2 p r - 2 7 t , 2 a ; l ) ,
(&j> ft)i,M, (&j» ft; Bj)M+i,Q
(dj, aji Aj)i<N, ( t t j , a j ) ^ + l i p
(l=sj=2ffi=?2t±sj(7;i) (4.4.3)
using (4.4.2), (4.4.3), from(4.4.1) one gets required cosine Fourier Series.
w-l T?E;F;F' (sin xr-lF§$%, te);W;M; fainx)*
xH P,Q M,N
2 (sin x) 2<7
(&/» ft)i,Af > (bj,(3f, BJ)M+I,Q
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Mathematics A. M. U. Aligarh
v frM,N+l {*?-pr-Tt,2a;l),
(bj,(3j)\,M, (bj, ft; SJ)M+I,Q
(a,, a,-; A/)I,.N> (aj> aj)iV+i,P
( i ^ _ 2 p r - 2 7 t ) , c r ; l )
00 00
+ E £ e e ' ^ cos px (a /4 ' ) r (/?/47)' 1
p=-oo r,£=0 t! '2w-2
x t f P+l .Q+2
( l - w - 2 p r - 2 7 t , 2 c r ; l ) ,
(bj, ft)i,Ar, (6j, ft; 5J)M+I,Q
(aj, otj\ Aj)itN, (aj, <XJ)N+I,P
( 1 - " - 2 ^ ± m! a - ; l )
(4.4.4)
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Applied Mathematics A. M. U. Aligarh
4.5. SINE FOURIER SERIES :
Let
f(x) = (sin xy'"lF^$ (g);(h);(h'); {3(sin x)2^
frP,Q z(sin x) 2a (oj, ay AJ)I,N, (dj, otj)N+hP
{bj, Pj)i,Mi (Pj, (5y Bj)M+hQ
= Yl Cp sin px. (4.5.1) p=—oo
Multiplying by eimx both sides in (4.5.1) and the integrating it with re
spect to x frome 0 to 7r ,and making to (4.1.11) and (4.2.1), one obtains
_ e*W2 ^ (a/4P)r (pl^t
r,t=0 *t *!
frM,JV+l x - n P + l , Q + 2 4"
(l-cj-2pr-27i,2(r;l),
(fy, /?J)I,M> (6j, ft; Bj)M+hQ
(aj, ay Aj\N, (aj, aj)N+hP
( l-u>-2pr-2-yt±m
, * ; i ) 4 .5.2)
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Now making an application of (4.5.1) and (4.5.2) we get required sine
Fourier Series.
wi-l TTE;F-F' (2 sin xr-lFffi%, (<0; (/);(/ '); <*(sm *)2" to); CO; (>*'); ftsmx)*
xHP'Q
xnM,N z(sin x) 2<T
(a,j, af, Aj)hN, (dj,aj)N+i,p
(bj, Pj)i,M, {bj, Pj] Bj)M+lQ
oo oo 2eeiP*/2 . (a/4P)r (p/VY E E : sin px e , ,, -
p=-oor,t=0 l T\ V.
x/7P+l,Q+2
(l-v-2pr-2jt,2cr;l),
(bj, (3j)i.M, [bj,fij] Bj)M+i,Q
(aj, ay, Aj)hN, (a,-, aj)N+hP
(4.5.3)
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4.6. MULTIPLE EXPONENTIAL F O U R I E R SERIES :
Let
f(xi • • •, xn) = (sin x)'"'1-1 • • • (sin x)1"^1
( e i ) ; ( / i ) ; ( / { ) ; « i ( « « i ) 2 f t "
( P I ) ; ( M ; W ) ; Pi(sinXl)**_
(en); (/n); (/„); an{sin xn)2Pn
(9n)\ (hn); {tin); Pn{sin Xn)2^
(aj> &j'i ^j)l,N, (%) (*J)N+1.P
(bj, (3j)i,M, (bj, fy; BJ)M+I,Q
30 00
= E ••• E A P l . . . P n e - ^ + - + ^ ) . (4.6.1) Pi=-oo pn=-oo
Equation (4.6.1) is valid, since f{x\, • • •, xn) is continuous and of bounded
variation in the open interval (0,7r). In the series (4.6.1), to calculate
APu...jn and fix xu • • •, ar„_i, so that
116
F, ^ l ^ l i - ^ l
Gi\Hi\H[
pEn;Fn;Fn
Gn;Hn;H'n
xH P.Q M,N
z(sin Xi)2°l (s in a;n)2<7n
Applied Mathematics A. M. U. Aligarh
oo oo V A p-*(Pl3;l+-+Pn-lXn_i) 2^ ^Pr -Pn- i^
Pl = -0O p n _ ! = - 0 0
depends only on pn.
Furthermore ,it must be the coefficient of Fourier exponential series in xn
of / (xi , • • •, xn) over 0 < xn < ir.
Now multiplying by elTTlnXn both sides in (4.6.1) and integrating with
respect to xn from 0 to 7r,one gets
(sin Xi) Wi-l (sin xn) wn-l
(ei); (/i); (/{); (^1);(/i1);K);
ai(sin x\)2pl
Pi(sin x{flx
jiEn-i;Fn-l',Fn-l Gn~\;Hn-\;H'n_x
( e n - l ) ; ( / n - l ) ; ( / n - l ) ; ^ n - l ^ m Z ^ ) 2 ' ' " - 1
( ^ n - l ) ; ( h n - l ) ; ( ^ n - l ) ; Pn-\{sin I n _ i ) 2 V l
v f^fain T \wn-lJmnxnpEn;Fn;F^ x yQ ^szn z n ; e rGn.Hn.H,n
(O; (/«); (/i); an(sm ^n)2^ Gfo); (&»);«); 0n(sin xn)
2^
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xffP'Q xnM,N
2a„ z(sin Xi)2(Tl • • • (sin xn)2(Tn
(a,-, oiji Aj)iiN, (a,, OCJ)N+I,P
(PJ, ft)i,Af. (6j, ft; BJ)M+I,Q
dxr
0 0 oo
= E ••• E iV^e-*0^--^*") pi=-oo p„_i=-oo
4-00 . ^
g l (C<("»n-Pn)xBrfa. p„=-oo
(4.6.2)
using(4.1.9) and (4.2.1), from (4.6.2), respectively, one gets
oo oo gi(pH \-pn)n/2
APv..Pn = E • • • E 2(a.,1+...+a;n)_„ (d, • • •, €n)
X ( a i / ^ r (ft/4^1
n! ti!
K/4^)r"(/V4>) r ! f I
tn
fjM,N+l xnP+hQ+l 4(CTJH hon)
(1 - W l - 2pm - 2 7 ^ , 2 ^ ; 1).
(1 - w„ - 2pnrra - 27„in, 2<rn; 1), (a,-, a;-; AJ\N, (a,-, aj).v+i,p
n-a;1-2p Iri-271 t1±mi l ) . . . / , l -^n-2pnrn-27n tn±mn ^ . -j\ (4.6.3)
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Using (4.6.3) in (4.6.1), one obtains required multiple exponential Fourier
series.
Let
(sin xi)™1-1 • • • (sin x ^ ' 1
xF, E;F;F' G;H;H'
(ci);(/i);(/J); a^sin xx)2^
(pi);(fci);W); Piisinx^
-pEn;Fn;Fn Gn\Hn\H'n
(en); (/n); U'n)\ an{sin xn)2Pn
{9n)\ {hn); {h'n)\ pn{sin xn)2^
xnM,N z(sin Xi)2<Tl • • • (sin xn)2(Tn
(a,-, ay, Aj)i^N, (aj, a ^ + ^ p
(fy> Pj)\,M-> (fy> % BJ)M+I,Q
00
E L Pl ," ,Pn = -3C ^ - r , , , * ! — t n = 0
(ei • • • e n ) 2(wi+--+w„)-n
x c - i < * » I + - + » .«n) e(p1 + -+P„)iV2 ( ^ l / ^ ) r i (ft/**)''
ri! ti!
(an/4^) rn 0^/4^) *n
*»!
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Applied Mathematics A. M. U. Ahgarh
frM,N+n xnP+n,Q+n 4(^1+ +On)
( l - w i - 2 p 1 r i - 2 7 l t 1 , 2 ( T i - > l )
{bj,Pj)i,M, (bj,Py,Bj)M+i,Q
(1 - u)n - 2pnrn - 2-yntn, 2an\ 1), (a,, (aj5 a/, A , ) ^ , {ava3)N+\,p
n-g i i -2p in -27 i^ i±Tn i a . ]\ (l-un-2pnrn-2intn±mn ^ . j \
(4-6.4)
4.7. P A R T I C U L A R CASES
Setting ft, •••,/?„ = 0 in (4.2.2), one gets L.H.S. of (4.7.1)
j * • • • I*(sin Xi)Wl-\ • • •, (szn x ^ " - 1 e
%(miTu 'WnT"
Ei+Fi^Gi+ffi
( c i ) ; ( / i ) ;
(01); ( M ;
ai(sin xi)2pi
En+Fn n
(e«);(/n);
(9n);{hn);
an(sm xn)2pn
YHP>Q
xnM,N z(sin Xi)2<Tl • • • (sin xn)
2°n {a3 <ij]Aj)itN,
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{a.j,aj)tf+itp
(bj,(3j;Bj)M+i,Q
dxi, • • • ,dxTl
(7{\n(,i(mi+--+mn)n/2 oo
9(wi+-+wn)-n ^ G, " Hi ~ r — ° n teUl)r, n ( S ) r i
K „ — 1 fCn — 1
frM,N+n x - n P+n,Q+2n 4 ( f f l + - +<7n)
( l - w i - a p i r i ^ f i ; ! ) . - .
(6j,/3j)l,M, {bj,PfBj)M+l,Q
•••{l-Un- 2^nrn, 2an; 1), (aJ5 a,; A,) l i iV , (aj, OCJ)N+L.P
Further setting a i , • • •, an — 0 in (4.7.1).
Next one gets L.H.S. of (4.7.2)
(4.7.1)
J* • • • / " ( s in z i f 1 " 1 • • • (sin Xn)^-iei(mlXl+-+mnxn
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xH M.N P,Q
z(sin z i ) 2 < V • •, (sin xn) 2<Tn
(7r)V( 7n14-"+rnn)ir/2
2(oJi+---+ai2)-n
{a,j,OLj\Aj)iji,
(fy;ft)i,M,
{aj,aj)N+hP
(bj,(3j-,Bj)M+i,Q
dx\•• • dxn
xH M.N+l P+n,Q+n 4(^1 + - •+an)
( l - w i , 2 c r r , l ) - - - ( l -w„,2(7„; l ) ,
(fy> ft)i,Af >(fy, ft; BJ)M+I,Q
(flj, oij-; i4J-)iijv, (flj, aj)jv+i,p
(4.7.2)
Now setting a = /? = 0 in (4.3.3) one establishes
(oj, o -; Aj)i,J\r» (ftj, <*J)N+I,P - 1 frM,N (sin i ) - 1 ^ z(sin x) 2(7
(&7>ft)i,M, (fy,ft; BJ)M+\,Q
00 p ip ( f -x )
r -— p=—00 ^
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xH M.N+l P+l,Q+2
(1 - a>, 2a; 1), (aj7 ay, Aj)hN, (ay <XJ)N+I,P
(bj, ft)i,M, {bj, ft; BJ)M+I,Q (^f^2, o-; l)
. (4.7.3)
Letting p = 2/ as 1 is a integer,from (4.7.3), one establishes:
L.H.S. of (7.3) = A'
X-"P+1,Q+1
(^ i ,a;l) ,(a J-,a J-;i4j)i ) jv,(ai,ajWi,p
(6j, ft)i,A/, (fy, ft; BJ)M+I,Q ( ^ , o"; 1)
1 (X)
52 COS /7T COS 2/X. Ow-2 ^ ,
£>Af,JV + l x - "P+ l ,Q+2
(1 - w/r; 1), (ai5 ay, A,-)I,JV, (a,-, a^jv+i.p
(fy, ft)i,A/> (feji ft; #J)M+I,Q (1~^:fc2i, <T; l)
(4.7.4)
Further letting p = (21 + 1) as 1 is an integer,from (4.7.3) one obtains
L.H.S. of (4.7.3)
1 OO
= — ^ Y, sin (21 + l)7r/2 . sin (21 + l)x 2 U l
p=i
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(1 - UJ, 2a; 1), (a,-, ay, Aj)hN, (a.,-, otj)N+i,p
{bj, /?J)I,M, {bj, Pj\ Bj)M+hQ ^1~^±2(
2<+1)) a ; i j
(4.7.5)
Similarly, remaining particular cases can be evaluated by (4.4.4) and
(4.5.3) applying the same techniques.
xH M,N+1 P+l,Q+2
z 4"
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CHAPTER V
Applied Mathematics A. M. U. Aligarh
CHAPTER V
INTEGRALS AND FOURIER SERIES
INVOLVING ff-FUNCTION
ABSTRACT: The present chapter deals with the evaluation of an in
tegral involving an exponential function, sine function, generalized hy-
pergeometric series and //-function and the same has been employed to
evaluate a double integral and establish Fourier series for the product
of generalized hypergeometric functions. A double Fourier exponential
series for the //-function has also been derived.
5.1. INTRODUCTION:
The //-function occuring in this chapter is defined and represented in
the following manner [18]:
(a,-, OCJ] Aj)hN, (aj, otj)N+lf
(bj, ft)i,M, (&j, ft; BJ)M+I£
rM.N f?S"W =
fjM,N HP,Q
Applied Mathematics A. M. U. Aligarh
where
M N
n r(bj - 0jO n {r(i - aj + ajz)}A' 4>(i) = - ^ ^ ? (5.1.2)
J Xi { r ( 1 -» , + ^ ) } B ' J . i} + . r ( a , - a ^ which contains fractional powers of some of the gamma functions.
Here, and throughout the chapter aj(j = 1, • • •, P) and bj(j = 1, • • •, Q)
are complex parameters, a3 > 0 (j = 1, • • •, P) , (5j > 0 (j — 1, • • •, Q)
(not all zero simultaneously) and the exponents Aj (j = 1, • • •, N) and
Bj (j — M + 1, • • •, Q) can take on non-integer values.
Buschman and Srivastava [18] has proved that the integral on the right
hand side of (5.1.1) is absolutely convergent when ft > 0 and | argz |<
- 7r 0., where 2
M N Q p
fi = E f t + E M - E PjBj- £ <*-,•>(). (5.1.3) j = l j = l j=M+l jf=JV+l
If one takes Aj = 1 {j = 1, • • •, N), Bj = 1 (j = m+ 1, • • • ,<£) in (5.1.1),
the function /fjsg^ reduces to the Fox's H-function.
The following formulae are required in the proofs:
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Mishra [108] has evaluated
f Jo
(sin x)w-leimx PFQ
ap ;c(sin x)
PQ
2h dx
7rei7»i7r/2 oo
O w - 1 *^> z r=0
(ap)rcrT{w + 2hr)
(pQ)rr\2^T ,w + 2hr±m+l\, (5.1.4)
where ap denotes e*i, • • •, ap; F(a ± b) represents T(a + b)T(a — 6); h is a
positive integer; P < Q {ov P — Q + 1 and \c\ < 1); no one of the PQ is
zero negative integer and Re (if) > 0.
The integral (5.1.4) can easily be established by expressing the hyperge-
ometric function in the integrand as [36 p. 181(1)] and interchanging the
order of integration and summation, which is justified due to the abso
lute convergence of the integral and the sum involved in the process, and
evaluating the inner integral [36 p. 12(29)].
Jo (sin x)w-leimx
PFQ ap ; c(sin x) \2h
U^V 7f/ ;d(sin x) 2fc
dx
ire nnx oo
- £ {aP)rcr (yu)t tfr(w + 2hr + 2kt)
•£o (PQ)r r\ (SV)t t\ 2^+kt)T fw+2hr+m±m+l\ ' (5.1.5)
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where in addition to the condition and notation of (5.1.5), k is a positive
integer; U < V (OT U — V + 1 and \d\ < 1); no one of 6y is zero or a
negative integer.
To derive (5.1.5), use the series representation for JJFV, interchange the
order of integration and summation and evaluate the resulting integral
with the help of (5.1.4).
The following orthogonality properties, can be established easily:
[* ei(m-n)x dx = t Jo
7r , m = n
Jo jimx cos nx dx =
7T
0
7T
2 7T
0
m = n = 0 m ^ n
m = n
m = n — 0 m ^ n
(5.1.6)
(5.1.7)
rir / etmx sin nx dec = < Jo
iri — , m = n (5.1.8) 0 , m ^ n
provided either both m and n are odd or both m and n are even integers.
In what follows for sake of brevity in addition to the notions earlier given
in this section, A and fi are positive numbers and the symbol (a ± 6, A)
represent (a + b, A), (a — 6, A), and
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H(x) = H% z(sin x) -2A
(fy) Pj)\,u, {bj, Pj\ Bj)u+^q
4>{r) = {ap)r cr
(PQ)r r\ ' m = {lu)t $
{Sv)t t\'
Hi(xy) = H% P,Q
z(s'm x) 2A(sin y) ~fl {aj7 ay, Aj)hv, (cij, aj)v+i,p
{bj,(3j)i,u, {bj, Pj] Bj)u+i,q
Fi(x) = PF( Q (ap) ;c(sin x)2h
(PQ) , F2{x) = uFv
7f/ ;d(sin x)2k
6v
5.2. T H E INTEGRAL :
The integral to be evaluated is
/ (sin x)w-leimxFl(x)F2{x) H{x)dx = yf{ir)eim^2 £ 0(r)V(t) r,*=0
x /7'w+2 '1 ' x -"p+2,9+2
(^•, ay; Aj)u,, (a,-, ar,-)«+i,p,
/tt;+2/tr+2fct ^ \ /w+2/tr+2fct+l \ \
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(5.2.1)
(w+2hr+2kt±m+\ \ \
(bj, /?j)l,u, (fy, Pj] Bj)u+i,q
provided that
£ dj - £ bj < 0, Re[w + 2A(1 - a^/bj] > 0, {j = 1, • • •, u).
| argz |< - 7r £7, ft > 0
together with the other conditions given in (5.1.4) and (5.1.5) are satisfied
[Q. is given by (5.1.3)].
P R O O F : To establish (5.2.1), express the //-function in the integrand
as the Mellin-Barnes type integral (5.1.1) and change the orders of inte
grations, which is justified due the absolute convergence of the integrals
in the process, to obtain
^ - T # 0 * * j O i n xr~2X'3-1eimxF1(x)F2(x)dxdC
Evaluating the inner-integral with the help of (5.1.6) and using multipli
cation formula for the Gamma-function [36; p.4(ll)], one gets
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Applied Mathematics A. M. U. Aligarh
w+2hr+2kt * )
V i j rh 27Ti J-ioo T ( " ± ^ + 2 ^ + 1 _ ^
xr w + 2hr + 2kt + 1
A^ *« #.
Now applying (5.1.1), the value of the integral (5.2.1) is obtained.
PARTICULAR CASES
(i) In (5.2.1), putting d = 0, it reduces to the form
fir — ^o
J (sin x)u'-leimxFl{x)H{x)dx = f^y™'2 £ <£(r) r=0
rrfc+2,J' -"p+2,0+2
( % , a i 5 ^ ) u „ (aj, aj)v+hp, ( " + ^ ± ™ + i , A)
( I t i 2- L , A), ( u + 2
r + 1 , A) , (6j, /3j)i,u, (6j, /?J; Bj)«+i,fl (5.2.2)
where the conditions of validity are same as stated in (5.2.1) with d = 0.
(ii) In (5.2.2), if one sets c = 0, it reduces to the following form
/"(sin x)w-leimxH{x)dx = v/(7r)eim7r/2
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frU+2,V x-"p+2,<7+2
[aj, otj] Aj)i^v, (aj, aj)v+\^Pi ( -^ , A]
(f > ^ ) ' \2~i ^ ) ' ft' ft)i,w ft' ft'5 Bj)u+l/I
(5.2.3)
where the conditions of validity are same as given in (5.2.2) with c = 0.
Separating real and imaginary parts in (5.2.3), one obtains
/ (sin x)w * cos mx H(x)dx = y^r) cos ra7r
fju+2,v x -"p+2,9+2
and
( a j , Q j ; Aj) i i l ( , ( a j , a j ) w + i j , , ( 2 1 )
if' ^J ' v ^ - ' J ' ft' ft)i,«> ft> ft 5 ^j)«+i,« (5.2.4)
/ (sin z)w * sin mxH(x)dx = )/(7r) sin mix
fjU+2,%) np+2,q+2
(a,, a* A,-)!,,., («j, «j)?;+1,p, (2±s±l, A)
(I'^J '04^'^) 'ft'ft)l,U' ft»ft"i^j)«+l, (5.2.5)
which follow same conditions as followed by (5.2.3).
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5.3. THE DOUBLE INTEGRAL
The double integral to be evaluated is
f f(sin x)Ul-1(sin y)w^1 ei{m' x+vi2y)
xFl{x)F2{x)Fl{y)F2{y) Hl{xy)dxdy
7re z(mi+m2)7r/2
00 00
£ £ ^1)^2)^1)^2) ri,ti=0r2,t2=0
TTU+4,V ( « i » " i ; ^ j ) i . r , ( ^ , a j ) r + l j p ,
/,u'i+2hn+2fcti ^ \ /w1+2/ir1+2fc^+l \ \
/'wi+2/tri+2A:tii:mi + l ^ fw2+2hri+2kt-i±Tni+l \
(5.3.1)
where in addition to the conditions stated in (5.2.1) Re[w;1 + 2A(1
ajloLj)] > 0, Re[w2 + 2/z(l - aj/ctj)] > 0, (j = 1, • • •, u).
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Applied Mathematics A. M. U. Aligarh
P R O O F : To establish (5.3.1), evaluate the x-integral of (5.3.1) with the
help of (5.2.1) and interchange the order of integration and summation,
to get
oo
0Fe im»'/2 £ 0 ( n ) 0 ( r 2 ) / (sin y r - ^ ^ F ^ y j F a f e ) r i , r2=0
xH. u+2,v p+2,q+2 z(sin y) - 2 / i
{a,j, aj; Aj)ifV, [dj, aj)v+ifP,
( wl+2hri+2kti
. * ) . ( wi+2/tn+2fc£i + l
^wi+2/tri+2fcti±mi+l \ \
, A ) ,
dy [bj, 0j)\,u{bj, Pj', Bj)u+i,q.
Now applying (5.2.1) to evaluating the y-integral, the value of (5.3.1) is
obtained.
PARTICULAR CASES :
(i) Putting d — 0 in (5.3.1), one gets
/ 0 ' ^ ( s i n x)-1-1(sin yr-1ei^x+m^Fl(x)F1(y)H1(xy)dxdy
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= 7re^mi+m9),r/2 E ^(n)^(r2) n,r2=0
x -rzp+4,g+4
( o ^ Q j i A ^ i ^ ^ a ^ Q l j ) ^ ! , P>
( m i | ^ i ) A ) , ( ^ + 22 ^ + i ) A ) )
/ ,w1+2fari±mi+l A"\ /w2+2/tr2i:m2+l .A
H P * , /*), (^±f*±±, fi) , (bjrfj)^, ft; B^+u (5.3.2)
where the conditions of validity are same as stated in (5.3.1) with d = 0.
(ii) Setting c = 0 in (3.2), one obtains
jTjTCsin x r ^ s m yT2-lei{miX+m^ Hl{xy)dxdy
i(mi+m2)ir/2 TJU+4,V ne -"p+4,q+4
(Oj, a^; A j ) i r , ( a j , a ^ + i j , ,
(f,A),(^,A),(f^)
(f.l').fe.ft)i,.ft1ftBiU (5.3.3)
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where the conditions of validity are same as in (5.3.2) with c = 0.
The integrals given in this section may be employed to establish dou
ble and multiple Fourier series for the product of hypergeometric functions
and the //-functions.
5.4. F O U R I E R SERIES :
The Fourier series to be established are
(sin x)w-lF1(x)F2(x)H{x)
1 oo oo _
E E mmH^y^1'1 ~x)-V 7T n = - o o r,t=0
(5.4.1)
(sin x)w-iFl{x)F2(x)H{x)
1 oo
E <t>(r)m yx r,t=o
TTU+1,V x np+l,q+l Z
(aj, ay, Aj)^v, {ajt aj)v+hp, (^ w+2hr+2kt+l • * )
( w+2hr+2kt
u+l ,
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Applied Mathematics A. M. U. Aligarh
O oo oo
^ = E E <P{r)4it)ein*/2H2{x)cosnx. V n n = l r,£=0
(5.4.2)
(sin x),,'_1Fi(a;)F2(a;)i/(a;)
2 °° °° -E E 0( r )V^)e W 2 £ 2 (x)s in na>.
V71"* n= l r.t=0 (5.4.3)
where n's are either even or odd in addition to the conditions of validity
followed by (5.2.1) and
u+2,v H2{x) = Hp+2,J+2
( a j ) Qj-; A j ) i ) 1 ; , ( a^ , a j ) r + i i P ,
Ar+2fcr+2fct \ \ /w+2/ir-f2fct+l \ \
fw+2/i,r+2fct±m+l \ \
{bj, Pj)i,u, (bj, (3j\ Bj)u+\.t
P R O O F : To prove (5.4.1), let
oo f{x) = (sin xr-'F^F^Hix) = £ A^ inx
n=—oo (5.4.4)
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The equation (5.4.4) is valid, since f(x) is continuous and bounded
variation in the interval (0, n).
Multiplying both sides of (5.4.4) by etmx and integrating with respect
to x from 0 to 7r, one gets
jT(sin x)w-leimxFl{x)F2{x)H{x) = f ) An £ J{m~n)x dx n=-oo
Now using (5.2.1) and (5.1.8), one gets
•Air, —
V*
00
e"/2 E 0(0 V>0 ft(m), r,t=0
where
(5.4.5)
u+2,v H2(m) = H%%+2
\a,j,a.j\ Aj)iA),
fw+2hr+2kt \\
(n r».\ , (w+2hr+2kt±m+l \\ \aj,aj)v+l,p, { 2 ' A)
Prom (5.4.4) and (5.4.5), the Fourier exponential series (5.4.1) is ob
tained.
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To establish (5.4.2), let
(sin x)"-1Fl{x)F2{x)H{x) = -£ + E £ucos ra. (5.4.6)
Multiplying both sides of (3.6) by elTnx and integrating with respect to x
from 0 to 7r and using (5.2.1) and (5.1.7), one gets
9 . oo
Bm = -n-eimw/2 £ <f>{rm)H2{m). (5.4.7) \J{n) r,t=o
From (5.4.6) and (5.4.7), the Fourier cosine series follows.
To prove (5.4.3) let
00 \w—l (sin x)u'-lFl{x)F2{x)H{x) = £ Cnsin nx. (5.4.8)
n=l
Multiply both sides of (4.8) by eimx and integrate with respect to x from
0 to 7r and use (5.2.1) and (5.1.8), to get
O 00
Cm = -n^e*"''2 £ tfW </>(*) #2(ro). (5.4.9) V(7T) r,<=0
From (5.4.8) and (5.4.9) the Fourier sine series is obtained.
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P A R T I C U L A R C A S E S :
(i) In (5.4.1), putting c = d = 0 and assuming n's as even integer, then
it is reduces to the form:
(sin x)w-lH(x) >/M
xH. u+l,v P+1,9+1
(aj, af, Aj)u„ (aj, OCJ)V+IJ» (w + 1, A)
(f, A), (bj, /?j)i,u, (bj,Pj] Bj)u+i,q
O oo
V(7I") n=l
r ru+2 ,v x - " p + 2 , 9 + 2
(aj, aj] Aj)itV, (aj, OLJ)V+\,P, [W ™+ , A)
(f, A), (^±i, A) , (bj, 0j)i,w {bj,Pj; Bj)u+iA
WK x cos — cos nx,
2 (5.4.10)
where the conditions of validity follows from (5.4.1).
(ii) In (5.4.1), putting c = d = 0 and assuming n's as odd integers, then
it yields the following Fourier series:
(sin x)w~lH(x) =
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TJU+2,V x - " p + 2 , g + 2
[aj, a.j\ Aj)ij,, (aj, otj)v+ip, ( ^ , AJ
n7r x sin — sin nx,
2 (5.4.11)
where the conditions of validity follow from (5.4.1).
Similarly the particular cases of (5.4.2) and (5.4.3) can be obtained.
5.5. DOUBLE FOURIER EXPONENTIAL SERIES :
The double exponential Fourier series to be established is
\w2~l 1 oo oo
(sin x r -^ s in y)**-lH,(xy) = - £ £ 3i(ni+n2)7r/2
7T J l l = —00 Tl2 = —OO
TTU+4,V
* •r2p+4,g+4
(aj, ctj\ Aj)i,v, (aj, aj)w+iiP,
(f^),(^,A),(f,M),
e ^ " 1 1 ^ ^ (5.5.1)
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where ii\'s are either odd or even integers and n2's are also either odd
or even integers; and other conditions of validity are same as stated in
(5.3.3).
P R O O F : To establish (5.5.1), let
f(x,y) = (sin x)w*-l(sm y^H^xy)
00 00
= E E Ani,n2e-^x+n^. (5.5.2) 7 l i = — OO Tl2= — 0 0
The equation (5.5.2) is valid, since f(x, y) is continuous and of bounded
variation in open interval (0, oo).
The series (5.5.2) is example of what is called a double Fourier expo
nential series. Instead of discussing the theory, we show one method to
calculate AnuTl2 from (5.5.2) is shown here. For fixed x, one notes that
00
7 l i = — OO
depends only on n2; furthermore, it must be the coefficient of Fourier
exponential series in y of f(x, y) over 0 < y < n.
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Multiplying both sides of (5.5.2) by eini2y and integrating with respect to
y from 0 to 7r, one obtains
(sin x)Wl-1 /"(sin y)w'~leirn2yHl{xy)dy
oo oo
= £ £ 41,n2e-'"17/!(ni-n2)^2/. /»!=—oo ri2=—oo
Now using (5.2.3) and (5.1.7), one gets
(sin xY"1'1 ,im2ir/2
xK u+2,v p+2,q+2 z(s'm x) \-2A
{aj^ Qj'i A.j)\j„ (a.j, Qljjtj+i^ ,
(f^),(^,//),
(&j, ft)i,u, (fy, ft; #j)u+i,.
V 4 p-ini^: ni=—oo
(5.5.3)
Multiplying both sides of (6.5.3) by elTniX and integrating with respect to
x from 0 to 7r and using (5.2.3) and (5.1.8), one gets
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Applied Mathematics A. M. U. Aligarh
A — _J{mi+Tn2)n/2TTU+4,v s*-m\,m2 c -t-'p+4,q+4
7T (f,A),(«5tl,A),(f,^),
\ 2 ' A) ' I 2 '^J
(^, / . ) , (6 i , /? J)1 ,u , (6 j , /? J ;5 J)u + 1 , ( 7
(5.5.4)
From (5.5.2) and (5.5.4), the double Fourier exponential series follows.
The double Fourier exponential series (5.5.1) can be derived with the help
of the integral (5.3.3), provided the theory of double Fourier exponential
series is developed on the lines of Carslaw and Jaeger [26.pp. 180-185]
Bajpai [13] and Singh and Garg [132].
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Applied Mathematics A. M. U.
CHAPTER VI
A FINITE INTEGRAL INVOLVING
GENERALIZED LAURICELLA'S
FUNCTION AND THE if-FUNCTION
WITH APPLICATION
ABSTRACT: The present chapter deals with a finite integral involving
generalized Lauricella function and the //"-function . This integral acts as
a key formula from which one can derive many useful particular cases. For
the sake of illustration two particular cases of the main results have been
obtained. At the end application of the main results have been given
by interconnecting them with the Riemann-Liouville type of fractional
integral operator.
Applied Mathematics A. M. U. Aligarh
6.1. I N T R O D U C T I O N :
The /f-function occuring in the chapter will be defined and repre
sented in the following manner [18]:
(a,-, af Aj)ijf, (aj, aj)ff+\tp
(&j> ft)i,M> (fy> ft; BJ)M+\,Q
^ P , Q H - #P ,Q
1 -too 27ri -/-ioo
(6.1.1;
where
M N nrf ty-Mnira-a i + a ^
«fl = - ^ ^ ? .J+i{r(i-6j. + ^)}^.j+ ir(a j-a^)
(6.1.2:
which contains fractional powers of some of the gamma functions.
Here, and throughout the chapter aj(j = 1, • • •, P) ,bj(j = 1, • • •, Q) are
complex parameters, a,- > 0 (j = 1, • • •, P) and /3j > 0 (j = 1, • • •, Q}
(not all zero simultaneously) and the exponents Aj (j = 1, • • •, N) and
Bj (j = M + 1, • • •, Q) can take on non-integer values.
Buschman and Srivastava [18] has proved that the integral on the
right hand side of (6.1.1) is absolutely convergent when Q > 0
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Applied Mathematics A. M. U. Aligarh
1 and | argz |< - n S7,
where
n= E ^ + E V i - E PjBj- £ a ,>0 . (6.1.3) j = l j=l j=M+l j=N+l
The behavior of the/f-function for small values of | z | follows easily from
a result recently given by Ruthic [ [124],p. 306 eq.(6.9)]:
H^[z\ = 0(| z \"), 7 = nfnm [ I t e^ / f t ) ] | z |-> 0 (6.1.4)
If one takes A,- (j = 1, • • •, JV) and £ , (j = M + 1, • • •, Q) in (7.1.1), the
function HPQ reduces to the fox's H-functions [20].
6.2. MAIN RESULT :
For the //-function of several complex variables defined by Srivastava and
Panda [156]
Here the following integral formula is derived
(V-d-*)'
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Applied Mathematics A. M. U. Aligarh
v jpv:A'; ;A^;0,Q *ra:B>; ;B<»);1,1 .)*-, i ^ t 1
' [ K M ' , ,6>W,A,A]:[(a'):^;
v [(&) :</>', ,^s),/i,/x]:[(6'):(5'];.
[(a(s)) : 0<*>];
[(&<">):#->]; [a+1;1]:[0+1;1];
\
Zx • • • Z s , —Xt, ( 1 — X) t
xH(zxh)dx
00
= £ n=o(a + l)„(/9 + l)„ n ( f tW,
' [(a, + r/A,) : 0', • • •, 0<->] :
v [(& + ^ ) : </>', • • •, </>(s)]
[(a '):^];---;[(a)-:0-]:
[(6'):5'];---;[(6)s:^] 21>" ' ' zai
148
Applied Mathematics A. M. U. Aligarh
^ ! nP+2,Q+2
(1 - e , / i ; l ) , ( l + e + a, / i ; l ) ,
(bj, ft)i,M, (&j» ft; BJ)M+I.Q,
(oj, Qj; >lj)ltjv, (flj, O:J)W+I,P
(1 - e -f a +- n, /i; 1), (/3 - e - n, /i; 1) (6.2.1)
where Re (/?) > - 1 , Re (e + h min fy^) > 0, j argz |< | 7r ft, ft >
0, [t |< 1 and the series on the right is convergent.
PROOF:
To prove (6.2.1), start with the following result Srivastava and Daoust
[143,p.l5,eq.(1.2)]
pf-A'; ;A^:0fi ro:B'; ;fl(s>:l,l
' IM:0\ ,9^\ A, A] :[(a') :0'];.
V Wo):^', , ^ ) , /z, /*]:[(&'): 6'}:
[(aW) : 0^];
[(&<->) :<5<'>]; [ a + 1 :1];[0 + 1:1]; z[ • • • z's, —xt, (1 - x)t
J
149
Applied Mathematics A. M. U. Aligarh
11 ((*j)n> 3=1
(a+ !)„(/?+!)„ II (ft) v:A'; ]A^ P°'p(l - 2r)Fu:A'; '^"'
rn Vx ^X)Cc:D'\ ;BW j = l
j M M j
/ [(a„ + nA„) : 0', ,6>W] : [(a') : </>'];
K^ + n^):^, ,^]:[(6'):<n;
••.[(aW):0W]; \ / • • • 7' in
where
Ai,0,-J), z = 1, -- • ,1/ , : j = l , . - . , s
and
(?)
and
and
& = 1 , • • • , # W
are all real and positive,(a,,) is taken to abbreviate the sequence of A^ pa
rameters oci • • • a,,, F(z[, • • •, z's) denoted the generalized Lauricella func
tion of s complex variables {z[, • • •, z's) Srivastava and Daoust [142,p.454].
150
Applied Mathematics A. M. U. Aligarh
Now multiplying both sides of (6.2.1) by xf~l(l — x)0 H[zxh) and
integrating if with respect to x from 0 to 1. Evaluating the right hand
side thus obtained, by interchange the order of integration and summation
(which is justified due to the absolute convergence of the integral involved
in the process ) and then interpreting the innerintegral with the help of
a known result Srivastava and Panda [156,p. 131' eq (2.21 )], one arrives
at the desired result.
6.3. PARTICULAR CASES :
(i) On specializing the parameters in a suitable manner and letting ax =
7, \i2 = S, Ai = A2 = 1. in (6.2.1), one obtains the following integral
£ x'-\l - xfH[zxh]F^ [7, < W • • •, w „ a + 1,
X (3 + 1; z[, • • • , z'a, -Xt.(l - x)t] dx
_^(l)n(S)n(-t)nr(p + n+l)
ntS (a + !)„(/? + l)nn!
151
Mathematics A. M. U. Aligarh
x Fj ' s )[7 + n , , <$ + n ; IUI , • • •, ws\ z[, • • •, 2.;]
x-"P+2,Q+2
( l - e , / i ; l ) , ( l + e + a,/i-,l),
(fy> ft)l,M, (fy» ft; BJ)M+1,Q,
(a,-, a,; Aj)i,N, (aj, aj)jv+i.p
(1 - € + a + n, /i; 1), (ft — e — n,h; 1)
valid under the same conditions obtainable from (6.2.1)
(6.3.1)
(ii) Putting z[, • • •, z's = 0 in (6.2.1), one obtains the following integral in
volving generalized Srivastava and Daoust function ([142],p. 199 and
[142],p.450)
fx'-^l-xfHlzx'
t/:0;0 qv:v;v [(<*,) : A, A] \
^ [(&):/u,/ i]:[a + 1 : 1 ] ; [ 0 + 1 : 1 ] ; — xt, (1 — x)t dt
I
152
Applied Mathematics A. M. U. Aligarh
n ( a , +nXj){-t)TT(/3 + n + l) J=I
r~o T{a + n+ l)r(/3 + n + 1) ft (ft + n/x ;>! E
v fjM,N+2 {l-e,h\l),{l + e + a,h\l),
(bj,Pj)i,Mi (fy> ft; BJ)M+I,Q,
(aj,otj;Aj)itN, (aj,otj)N+if
(1 - e + a + n, h\ 1), (0 - e - n, /i; 1) (6.3.2)
valid under the same conditions obtainable from (6.2.1).
6.4. APPLICATION :
The Riemann-Liouville fractional derivative of a function of order
a (for alternative, (-cr)th order fractional integral) [37,p.l81] is defined
here as follows:
«D*{/(*)} = rf^^(x-t)^'lf(t)dt .Re(a)<0.
di
(6.4.1)
I fea£W(s)},(g-l) < fie(a)<g,
153
Applied Mathematics A. M. U. Aligarh
where q is a positive integer and the integral exists.For simplicity, the
special case of the fractional derivative operator aD£ when a=0 will be
written as D£. Thus
Now by setting l=x in the main integral (6.2.1), it can be rewritten as
the following fractional integral formula
D^-11 (
{t f - i pV-A'y M<*>;0;0 ^V-B'; ;Bf');l;l
V
' [ K ) : 0 ' , ><),A,A] :[(<*'):</>'];•
• ; [ ( a W ) : ^ ) ] ;
.;[(6W):(5W]; [ « + ! : ! ] ; [5 + 1 : 1 ] ;
\
z[ -. • z'„-yt{l - t)y
\
J
x #,(***)}
I DO J = l
r(^ + i)n=o(a + i ) n ( / ? + i ) n n ( ^ ) r ^ J = I
154
Applied Mathematics A. M. U. Aligarh
A'- • A^ a- B'\ ;flW
(-t)nr(0 + n + l) n\
IW'.M,--
[ ( b ' ) - . S l ' -
•, l ( a j ) ' •• <f>s\ ••
z'v •, [(bjY : 5'] :
\
1 ^S
I
fjM,N+2 *nP+2,Q+2
(1 - e , / i ; l ) , ( l + e + a , / i ; l ) ,
(fy, ft)i,M, (fy> ft; BJ)M+I,Q,
(aj, a,-; Aj)hN, (a^, aj^+i^p
(1 - e + a + n, /i; 1), (/? - e - n, h; 1) (6.4.3)
where Re (77) > 0 and all the conditions of validity mentioned with (6.2.1)
are satisfied.
155
Applied Mathematics A. M. U. Aligarh
On specializing the parameters, generalized Lauricella's function and
the H-function of several complex variables may be transformed into G-
functions, E-functions, Lauricella's functions, Appell's functions, Kampe
de Feriet. functions, Bessel functions, and several other higher tran
scendental functions in one or more arguments. Therefore, the integral
transformations for various other functions of one or more variables can
be obtained as special cases of these results.
156
Applied Mathematics A. M. U. Aligarh
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