ON SOME PROBLEMS OF SPECIAL FUNCTIONS

T H E S I S SUBMITTED FOR THE AWARD OF THE DEGREE OF

Boctor of pi}iIo!^Qpl)p IN

APPLIED MATHEMATICS

BY

MUKESH PAL SINGH

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 ALlGARH-202002, U.P., INDIA

2004

SYNOPSIS

The present thesis embodies the researches carried out at the

Ahgarh MusHm University, Ahgarh. The thesis comprises of six chap­

ters. Besides the introductory I Chapter, the next chapter deals with a

study of a newly defined two-variables polynomial P„ /^(a:,^) which may

be regarded as a two-variables analogue of Legendre's polynomials P„(x).

The following chapters is a study of the calculus of T/.,T. ^-operator, a two

variable analogue of the operators studied earlier by W.A. Al-Salam

[2], H.B. Mittal [121-124] and M.A. Khan [50], [55], [60] and [69]. The

next two chapters are devoted to studies of new classes of polynomials

M^{x) and L'^"'^^'^'^'^'''\x,y,z) respectively. The last chapter concerns

an apphcation of special functions to Fluid Mechanics. In it a solution

involving hypergeometric function o^i has been obtained for miscible

fluid flow through a porous media.

Chapter I gives a brief survey of some of the earher and recent works

connected with the present thesis in the field of various hypergeometric

and double and triple hypergeometric type polynomials. The chapter

also contains some preliminary concepts and important well-known re­

sults needed in the subsequent text. An attempt has also been made in

this chapter to make the present thesis self contained. Further a brief

chapter wise summary of chapter II through VI have also be given in

this introductory chapter.

Chapter II deals with a study of a two variables polynomial P„^ (a:, y)

analogous to Legendre polynomial Pr,{x). The chapter contains differen­

tial recurrence relations, a partial differential equation, double generat­

ing functions, double and triple hypergeometric forms, a special property

and a bilinear double generating functions for the newly defined two vari­

ables polynomial.

Chapter III concerns with a study of the calculus of Tk r,,y-operator

The operator, which is a two-variable analogue of the operators given

earlier by W.A. Al-Salam [2] and H.B. Mittal [121-124] has been

used to obtain operational representations for various polynomials of

two variables which are useful in finding the generating functions and

recurrence relations of the two variable polynomials.

Chapter IV introduces a study of a new class of polynomial set M!^{x)

suggested by the polynomial A'^{x) of M.A. Khan and A.R. Khan

[93] and defined by means of generating functions of the form G{4x'^t —

QxH'^ + ixt^ ~ t^) for the choice G{u) = (1 — M)"^. The chapter contains

some interesting results in the form of recurrence relations, generating

functions, triple hypergeometric forms, integral representation, Schlafli"s

Contour Integral and Laplace Transforms.

Chapter V deals with a study of general class of polynomial

L|,''*'' ' -' '' '' ^(.T,?/, z) of three variables which include as particular cases

the various polynomials of one, two and three variables studied earlier

by a number of Mathematicians. Prominent among them are polynomi­

als L^^f\x) due to Prabhakar and Rekha [133-134], Ll^'^^^\x, y) due to

S.F. Ragab [135], Z^{x] k) due to Konhauser [112-113], U^{x) due to

Laguerre, Z!^-^^''^{x,y, z;k^p,q) due to M.A. Khan and A.R. Khan

[100] and three and m-variables analogues of Laguerre polynomials due

to M.A. Khan and A.K. Shukla [73-74]. The chapter contains cer­

tain generating functions, a finite sum property, integral representations

Schlafli's contour integral, fractional integrals, Laplace transforms and a

series relation for this general class of polynomials.

In chapter VI, a solution involving hypergeometric function oFi has

been obtained for miscible fluid flow through a porous media. Here, the

phenomenon of longitudinal dispersion has been discussed, considering

the cross-sectional flow velocity as time dependent in a specific form.

Ill

ON SOME PROBLEMS OF SPECIAL FUNCTIONS

T H E S I S SUBMITTED FOR THE AWARD OF THE DEGREE OF

Bottor of |pi)tIo£(opi)p IN

APPLIED MATHEMATICS

BY

MUKESH PAL SINGH

Uoder the Sopervision 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

2004

T6650

^r«^ tff;^_t_

PRAYER

I begin in the name of Almighty God, the Most Gracious, the

Most Merciful and the Sustainer of the worlds. All praise be to the

Almighty God without whose will this thesis would not have seen

the daylight.

The primary aim of education is to prepare and qualify the

new generation for participation in the cultural development of the

people, so that the succeeding generations may benefit from the

knowledge and experience of their predecessors.

DEDICATED

TO MY

ESTEEMED MOTHER

MRS. RA^AVATl DEVI

&

IN THE LOVING MEMORY

OF MY

FATHER

LATE MR. MANOHAR SINGH

TO

MY BELOVED WIFE

MRS. GEETU SINGH

&

MY DEAR CHILDREN

KSHATIJ PRATAP SINGH (SON)

MISS MAMTA SINGH (DAUGHTER)

MISS NISHU SINGH (DAUGHTERl^

DEPARTMENT OF APPLIED MA^l'HEMATICS, ZAKIR HUSAIN COLLEGE OF ENGGINEERING & TECHNOLOGY FACULTY OF ENGINEERING, ALIGARH MUSLIM UNIVERSITY, ALIGARH - 202 002, U.P. INDIA

CERTIFICATE

CERTIFIED THAT THE THESIS ENTITLED ''ON SOME

PROBLEMS OF SPECIAL FUNCTIONS" IS BASED ON A PART OF

RESEARCH WORK MR. MUKESH PAL SINGH CARRIED OUT

UNDER MY GUIDANCE AND SUPERVISION IN THE

DEPARTMENT OF APPLIED MATHEMATICS, FACULTY OF

ENGINEERING, ALIGARH MUSLIM UNIVERSITY, ALIGARH. TO

THE BEST OF MY KNOWLEDGE AND BELIEF THE WORK

INCLUDED IN 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. MUKESH PAL

SINGH HAS FULFILLED THE PRESCRIBED CONDITIONS OF

DURATION AND NATURE GIVEN IN THE STATUTES AND

ORDINANCES OF ALIGARH MUSLIM UNIVERSITY, ALIGARH.

(PROF. MUNTAZ AHMAD KHAN) SVPERVISOR

DATED: 1^0-L-^^^h

Dopartmont of .Applied MathenmHci O t CoIleoQ of Engg. & Tech.. AMU, Allgai*

CONTENTS

ACKNOWLEDGEMENT i v

PREFACE i-iii

CHAPTER I : INTRODUCTION 1 23

1 : SPECIAL FUNCTION 1

AND ITS GROWTH

2 : ORTHOGONAL POLYNOMIAL 3

3 : DEFINITION NOTATION AND 5 RESULTS USED

4 : A STUDY OF TWO VARIABLES 15 LEGENDRE POLYNOMIALS

5 : A NOTE ON A CALCULUS 16 FOR THE T^,,.,y-OPERATOR

6 : ON A NEW CLASS OF POLYNOMIALS 18 SET SUGGESTED BY THE POLYNO­MIAL OF MA. KHAN AND A.R. KHAN

7 : ON A GENERAL CLASS OF POLYNO- 19 MIAL OF THREE VARIABLES

8 : SOLUTION OF MISCIBLE FLUID FLOW 18 THROUGH POROUS MEDIA INTERMS HYPERGEOMETRIC FUNCTION o^i

CHAPTER II : A STUDY OF TWO VARIABLES 24-4

LEGENDRE POLYNOMIALS

1 : INTRODUCTION 24

2 : DOUBLE GENERATING FUNCTION 25 OF THE FORM G{2xs - s , 2yt - t^)

3 : THE LEGENDRE POLYNOMIALS 28 TWO VARIABLES

4 : DIFFERENTIAL RECURRENCE 32 RELATION

5 : PARTIAL DIFFERENTIAL EQUATION 35 0FP„4x,y)

6 : ADDITIONAL DOUBLE GENERATING 37 FUNCTIONS

7 : TRIPLE HYPERGEOMETRIC FORMS 40 OFP,,,{x,y)

8 : A SPECIAL PROPERTY OF 43

9 : MORE GENERATING FUNCTIONS 44

CHAPTER III : A NOTE ON A CALCULUS 46 5 FOR THE 71^,j/-OPERATOR

1 : INTRODUCTION 46

ACKNOWLEDGEMENT

Words and lexicons can not do full justice in expressing a deep sense

of gratitude and indebtedness which flows from my heart towards my

esteemed supervisor PROFESSOR DR. MUMTAZ AHAMD KHAN,

Department of Applied Mathematics, Faculty of Engineering, Aligarh

Muslim University, Aligarh. Under his benevolence and able guidance, I

learned and ventured to write. His inspiring benefaction bestowed upon me

so generously and his unsparing help and precious advice have gone into the

preparation of this thesis. He has been a beckon all long to enable me to

present this work. I shall never be able to repay for all that I have gained

from him.

I shall be always grateful to Ex-Chairman Prof. Tariq Aziz and

Prof. Aqeel Ahmad and to present Chairman Prof. Mohd. Mukhtar AH,

Department of Applied Mathematics, Faculty of Engineering, Aligarh

Muslim University, Aligarh for providing me necessary research facilities in

the department.

I am particularly indebted to Dr. Arjun Singh (Sr.ENT Surgen)

District Hospital Aligarh, Dr. Har Veer Singh (Ex-Chairman, Department

of Miltary Science, D.S. College, Aligarh), Mr. Vijay Vishal (Government

Contactor and builders) Mr. Rakesh Sharma Bohre and, Mr. Grish

Tomar and my cousins Mr. S.P. Singh, Mr. Harendra Kumar Singh, Dr.

Y.P.Singh, Mr. Rajesh pal Singh, Mr.Satendra Pal singh for their

inspiration and encouragement.

I also take this opportunity to thank the following staff members of

the Department of Applied Mathematics who have contributed in any way

towards completion of this task.

READERS:

1. Dr. Abrar Ahmad Khan

2. Dr. Arman Khan

3. Dr. Merajuddin

4. Dr. Aijaz Ahmad Khan

5. Dr. Mohd. Salim

LECTURERS

6. Dr. Abdul Hakim Khan

7. Dr. Mohd. Din Khan

8. Dr. Rafiquddin

9. Dr. Mohd. Kamarujjama

10. Dr. Shamshad Hussain

11. Mr. Mohd. Swaleh

SUPPORTING STAFF

12.Mrs.Niswan Ali

13. Mr. Sirajul Hasan (U.D.C.)

14. Mr. Mohd. Tariq

15. Mr. Abdul Viqar Khan

16. Mr. Mohd. Amin Khan

17. Mr. Ahmad Ali

18. Mr. S. IshtiaqAH

19. Mr. Mohd. Siddiq

20. Mr. Khalid Hussain

My thanks are also due to my seniors Dr. Mrs. Muqaddas Najmi,

Reader, Women's Polytechnic, A.M.U., Aligarh, Dr. Abdul Hakeem

Khan, Senior Lecturer, Department of Applied Mathematics, A.M.U.,

Aligarh, Dr. Ajay Kumar Shukla, Lecturer, Regional Engineering College,

Surat( Gujrat), Dr. Khursheed Ahmad, Trained Graduate Teacher, S.T.

High School, A.M.U.,Aligarh, Dr. Ghazi Salama Mohd. Abukhammash,

West Bank, Ghaza Strip, Palestine, Dr. Abdul Rahman Khan Lecturer,

Bijnor and Mr. Bhagwat Swaroop Sharma (Ph.D. thesis submitted), all of

whom obtained their Ph.D.Degrees under the supervision of my Supervisor,

for their help and encouragement in the completion of my Ph.D. work. I am

also thanksful to my other seniors Mr. Syed Mohd. Amin, P.G.T., city High

School, A.M.U., Aligarh, Mr. Ajeet Kumar Sharma, Moradabad,

Mr.Shakeel Ahmad Alvi, P.G.T., Senior Secondary School,

A.M.U.,Aligarh, Dr. Khursheed Ahmad, T.G.T., ST., High School,

A.M.U.,Aligarh, all of whom obtained their M.Phil. Degrees under the

supervision of my supervisor Prof. Mumtaz Ahmad Khan, for their help

and co-operation rendered in writing my Ph.D. thesis.

I wish also to thank my fellow research worker, Mr. Islam Khan,

Mr. Adil-Ur-Rahman and Mr. Naseem Ahmad Khan, for the pains they

took in going through the manuscript of this thesis for pointing out errors

and omissions in it.

I shall be failing in my duties if I do not express my gratitude to the

wife of my supervisor Mrs. Aisha Bano (M.Sc. B.Ed), T.G.T., Zakir

Hussain Model Senior Secondary School, Aligarh for her kind hospitality

and prayer to god for my welfare.

I am also thankful to my supervisor's children Miss Ghazala Perveen

Khan (daughter) B.Tech. (Electrical), Mr. Rashid Imran Ahmad Khan

(Son), B.Tech. Iir " Year (Petro-Chemical), student of Z.H. College of

Engineering and Technology, A.M.U. Aligarh , and to 11 years old Master

Abdul Razzaq Khan(son), a class VI student of Our Lady Fatima School,

Aligarh for enduring the preoccupation of their father with this work.

I thankfully acknowledge the co-operation rendered by my younger

brother Mr. Birjesh Pal Singh, his wife Mrs. Babli Singh and their

children.

I also express my deep appreciation to my beloved wife Mrs. Geetu

Singh and to my dear children Kshatij Pratap Singh (Son) and Mamata

Singh and Nishu Singh (daughters) and Master Raman Pratap Singh

(Ranu) and Master Sksham (Chotu) for bearing with me my preoccupation

with this thesis.

In the end I would like to thank Mr. Hasan Naqvi for his painstaking

and excellent computer typing of the manuscript of my Ph.D. thesis.

Finally, I owe a deep sense of gratitude to the authorities of Aligarh

Muslim University, my Alma mater, for providing me adequate research

facilities.

(MWCESH PAL SINGH)

DEPARTMENT OF APPLIED MATHEMATICS ZAKIR HUSSAIN COLLEGE OF ENGINEERING & TECHNOLOGY FACULTY OF ENGINEERING ALIGARH MUSLIM UNIVERSITY, ALIGARH-202002, (U.P), INDIA.

PREFACE

The present thesis embodies the researches carried out at the

Aligarh Muslim University, Ahgarh. The thesis comprises of six chap­

ters. Besides the introductory I Chapter, the next chapter deals with a

study of a newly defined two-variables polynomial P„^k{x,y) which may

be regarded as a two-variables analogue of Legendre's polynomials Pn{x).

The chapter that follows is a study of the calculus of T^^r,7/-operator, a

two variable analogue of the operators studied earlier by W.A. Al-

Salam [2], H.B. Mittal [121-124] and M.A. Khan [50], [55], [60] and

[69]. The next two chapters are devoted to studies of new classes of

polynomials M^(x) and L^"'^'^'^'^'^'\x,y,z) respectively. The last chap­

ter concerns an application of special functions to Fluid Mechanics. In

it a solution involving hypergeometric function o i has been obtained

for miscible fluid flow through a porous media.

Chapter I gives a brief survey of some of the earlier and recent works

connected with the present thesis in the field of various hypergeometric

and double and triple hypergeometric type polynomials. The chapter

also contains some preliminary concepts and important well-known re­

sults needed in the subsequent text. An attempt has also been made in

this chapter to make the present thesis self contained. Further a brief

chapter wise summary of chapter 11 through VI have also be given in

this introductory chapter

Chapter II deals with a study of a two variables polynomial P„ k [x. y)

analogous to Legendre polynomial P„ [x). The chapter contains differen­

tial recurrence relations, a partial differential equation, double generat­

ing functions, double and triple hypergeometric forms, a special property

and a bilinear double generating functions for the newly defined two vari­

ables polynomial.

Chapter III concerns with a study of the calculus of T/ .,^,^-operator.

The operator, which is a two-variable analogue of the operators given

earlier by W.A. Al-Salam [2] and H.B. Mi t ta l [121-124] has been

used to obtain operational representations for various polynomials of

two variables which are useful in finding the generating functions and

recurrence relations of the two variable polynomials.

Chapter IV introduces a study of a new class of polynomial set M'/,{x)

suggested by the polynomial A'^,{oi) of M.A. K h a n and A.R. Khan

[93] and defined by means of generating functions of the form G(4a:' t —

Qx'^t^ + ixt'^ — f^) for the choice G{u) = (1 — w)^^. The chapter contains

some interesting results in the form of recurrence relations, generating

functions, triple hypergeometric forms, integral representation, Schlafli's

Contour Integral and Laplace Transforms.

Chapter V deals with a study of general class of polynomial

L':^''^'^^ ^''\x^y,z) of three variables which include as particular cases

the various polynomials of one, two and three variables studied earlier

by a number of Mathematicians. Prominent among them are polynomi­

als L^'/ix) due to Prabhakar and Rekha [133-134], L^'^l^\x,y) due to

S.F. Ragab [135], Z;^(x; k) due to Konhauser [112-113], U^{x) due to

Laguerre, Zl^-'^'^'{x,y,z\k,'p,q) due to M.A. Khan and A.R. Khan

[100] and three and m-variables analogues of Laguerre polynomials due

to M.A, Khan and A.K. Shukla [73-74]. The chapter contains cer­

tain generating functions, a finite sum property, integral representations

Schlafli's contour integral, fractional integrals, Laplace transforms and a

series relation for this general class of polynomials.

In chapter VI, a solution involving hypergeometric function ()F\ has

been obtained for miscible fluid flow through a porous media. Here, the

phenomenon of longitudinal dispersion has been discussed, considering

the cross-sectional flow velocity as time dependent in a specific form.

Ill

CHAPTER I

INTRODUCTION

1. SPECIAL FUNCTIONS AND ITS GROWTH: A special func

tion is a real or complex valued function of one or more real or com­

plex variables which is specified so completely that is numerical values

could in principle be tabulated. Besides elementary functions such as

x", e \ logx, and sinx, "higher functions'', both transcendental (such as

Bessel functions) and algebraic (such as various polynomials) come under

the category of special functions. The study of special functions grew

up with the calculus and is consequently one of the oldest branches of

analysis. It flourished in the nineteenth century as part of the theory of

complex variables. In the second half of the twentieth century it has re­

ceived a new impetus from a connection with Lie groups and a connection

with averages of elementary functions. The history of special functions

is closely tied to the problem of terrestrial and celestial mechanics that

were solved in the eighteenth and nineteenth centuries, the bounadry-

value problems of electromagnetism and heat in the nineteenth, and the

eigenvalue problems of quantum mechanics in the twentieth.

Seventeenth-century England was the birthplace of special functions.

John Walhs at Oxford took two first steps towards the theory of the

gamma function long before Euler reached it. Wallis had also the first

encounter with elleptic integrals while using Cavalieri's primitive fore­

runner of the calculus. [It is curious that two kinds of special functions

encountered in the seventeenth century, Wallis elleptic integral and New­

ton's elementary symmetric functions, belongs to the class of hypergeo-

metric functions of several variables, which was not studied systematically

nor even defined formally until the end of nineteenth century]. A more

sophisticated calculus, which was made possible the real flowering of spe­

cial functions, was developed by Newton at Cambridge and by Leibnitz

in Germany during the period 1665-1685. Taylor's theorem was found by

Scottish mathemtician Gregory in 1670, although it was not published

until 1715 after rediscovery by Taylor.

In 1703 James Bernoulh solved a differential equation by an infinite

series which would now be called the series representation of a Bessel

function. Although Bessel functions were met by Euler and others in var­

ious mechanics problems, no systematic study of the functions was made

until 1824, and the principal achievements in the eighteenth century were

the gamma function and the theory of elleptic integrals. Euler found

most of the major properties of the gamma function around 1730. In

1772 Euler evaluated the Beta-function integral in terms of the gamma

function. Only the duplication and multiplication theorems remained

to be discovered by Legendre and Gauss, respectively, early in the next

century. Other significant developments were the discovery of Vander-

monde's theorem in 1772 and the definition of Legenedre polynomials

and the discovery of their addition theorem theorem by Laplace and Leg­

enedre during 1782-1785. In a slightly different form the polynomials had

already been by Liouville in 1772.

The golden age of special functions, which was centered in the nine­

teenth century German and France, was the result of developments in

both mathematics and physics: the theory of analytic functions of a com­

plex variable on one hand, and on the other hand, the theories of physics

(e.g. heat and electromagnetism) which required solutions of partial dif­

ferential equations containing the Laplacian operator. The discovery of

elleptic functions (the inverse of elleptic integrals) and their property of

double periodicity was published by Abel in 1827. Elleptic functions grew

up in symbiosis with the general theory of analytic functions and flour­

ished throughout the nineteenth century, specially in the hands of Jacobi

and Weierstrass.

Another major development was the theory of hypcrgeometric series

which began in a systematic way (although some important results had

been found by Euler and Pfaff) with Gauss's memoir on the 2F1 series in

1812, a memoir which was a landmark also on the path towards rigor in

mathematics. The 3F2 series was studied by Clausen (1828) and the iFi

series by Kummer (1836). The functions which Bessel considered in his

memoir of 1824 are o- 'i series; Bessel vstarted from a problem in orbital

mechanics, but the functions have found a place in every branch of mathe­

matical physics. Near the end of the century Appell (1880) introduced hy-

pergeometric functions of two variables, and Lauricella generalized them

to several variables in 1893.

The subject was considered to be part of pure mathematics in 1900,

applied mathematics in 1950. In physical science special functions gained

added importance as solutions of the Schrodinger equation of quantum

mechanics, but there were important developments of a purely mathe­

matical natuie also. In 1907 Barnes used gamma function to develop

a new theory of Gauss's hypergeometric function 2F\ Various genearl-

izations of 2F1 were introduced by Horn, Kampe de Feriet, MacRobert,

and Meijer. From another new viewpoint, that of a differential differ­

ence equation discussed much earher for polynomials by Appell (1880),

Truesdell (1948) made a partly successful effort at unification by fitting

a number of special functions into a single framework.

2. 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 apphcations to physics and to probability

and statistics and other branches of mathematics, the subject flourished

through the first third of this century. Perheps 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 apphcations 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

elleptic and associated functions. This drawback was overcome by E.

Heine through the definition of a generalized basic hypergeometric series.

3. DEFINITION NOTATIONS A N D RESULTS USED: Frequently

occuring definitions, notations and results used in this thesis are as given

under:

THE GAMMA FUNCTION: The gamma function is defined as

Viz

CO -, ,

f-^ e~' dt, Rl{z) > 0 ./o

T{z + l) Rl{z)<0, Zy^ 0,-1,-2,-3,

(3.i;

POCHHAMMER'S SYMBOL A N D THE FACTORIAL

FUNCTION: The Pochhammer symbol (A)„ is defined as

' A(A + l)(A + 2)---(A + n - l ) , if n =1,2,3,-••

(A). = I (3.2 f , n = 0

Since (f )„ = n\, (A)„ may be looked upon as a generalization of elemen­

tary factorial. In terms of gamma function, we have

(3.3) {A)„ = ^ . A ^ 0 , - 1 . - 2 , . . .

The binomial coefficient may now be expressed as

'A\ A ( A - l ) - - - ( A - - n + l) ( -1)"(-A),

.n m n\ (3.4)

Also we have

(A). = ( 1 - A )

Equation (3.3) also yields

, n = l , 2 ,3 , - - - , A: ;^0 ,±l ,±2 , (3.5)

W,n+v = (A)„,(A + m)„ (3.6)

which is conjunction with (3.5) gives

-1)"(A) (A) n — k 0<k<n.

( 1 - A - n)k

for A — 1, we have

i (-l)^r?!

n h = < (n-k)]

0 ,k > n

(3.7)

LEGENDRE'S DUPLICATION FORMULA: In view of the defi

nition (3.2), we have

w - ^ ^ i ^ K ^ l " "• • •' (3.

which follows also from Legendre's duplication formula for the Gamma

function, viz

7rr(22) = 2''~'r(z)r (z + ^-], ^ = ^ 0 , - ^ , - 1 , - ^ (3.9)

GAUSS'S MULTIPLICATION THEOREM: For very positive in­

teger, we have

III / x ^ ^ ^ i ^

{X)rr,r, = m ' " " J I - ^ ± ) ^ = 0 , 1, 2 , , J = l m

(3.10)

which reduces to (3.8) when m = 2.

THE BETA FUNCTION: The Beta function B{a,p) is a function of

two complex variables a and /3, defined by

Bia,f3) =

)ft-i r-\l - tf-'dt,Rl{a) > 0, Rl(P) > 0

r(^)r(/3) , I > + /5)

Rl{a) < 0, i? (/5) < 0. a, /3 / - 1 , - 2 , - 3

(3.11

THE GENERALIZED HYPERGEOMETRIC FUNCTION: The

generalized hypergeometric function is defined as

Qi,a2,- • • ,(ir

P],P2,- • • j(3q ;

oo

= E ai)n((y2] CXp jn Z

=0{Pl)ndP2)r,.r---iPq)r, n\

where/3, ^ 0 , - 1 . - 2 . - 3 , •••; j = 1, 2, • • •, (?.

The series in (3.12)

(3.12)

i) Converges for \ z \< oo iip < q.

ii) Converges for | z |< 1 if p = g + 1, and

iii) Diverges for all 2, 2 7 0, if p > ^ + 1.

Furthermore, if we set

then the pFq series, with p = q + 1, is

(I) Absolute convergent for | 2 |= 1, if Re(a;) > 0,

(II) Conditionally convergent for | ^ |= 1, z 7 1, If — 1 < Re(cL') < 0, and

(III) Diverges for | 2 |= 1 if Re(cj) < — 1.

Further, a symbols of type A(A;, cy) stands for the set of k parameters

Qcc + l cy -\- k ~ 1

V k k

Thus

p+k-t'q+r

a,j),A{k,a) ; z

- E - ( a i ) . ( a 2 ) „ - - - ( a p ) „ ( f ) J ^ ; „ , ,v .

r Jn \ r !n ^ 3 n\ (6l)r , (62)71 • • • (&g)«

in terms of hypergeometric function, we have ' V k /v

(3.13)

1 - 2 : °° 'a)„z" E F{)\a\_ \z

n\ \rMa.

00 ^r^

77=0 " • '

(3.14)

(3.15)

9

The linear transformations of the hypergeomctric function, known as

Euler's transformations are as follows:

2Fi[a,6; c; z] = {I - zY'\F, a,c — b] c; z z - 1

(3.16)

C 7 ^ 0 , - l , - 2 , - - - , | a r g ( l - z ) |< TT;

2Fi[a,6; c; z\ = {\ ~ z)'-"^\Fi\,c-a,c-h; c; z\ (3.17)

c ^ 0, - 1 , - 2 , • • • , I arg(l - z) |< TT.

Appell function of two variables are defined as

CO OO

Fi[a,b,b; q x,y\ = L L —TT-TTT:^ ^ 1/^ ^o«=o m! 72.'(c)„, +n

max {I ^ U 2/1} < 1;

(3.18;

CO OO

F2[a,6,6^ c^c'; x^y] = E E # ^ # 4 ^ ^ '"?/", , „ ^ 0 „ ^ 0 ^ ' Z ' (c)m (C

a: I + I T/ |< 1;

F3[a,a',b,b'; c; a:,i/] OO OO

max{ I a; I, I ?/ )} < 1;

3.19)

(3.20)

F4[a, b; c,c; x,y\ = l^ } ^ _ i ^i (^\ f ,V ^ ^ ' m- ^Or7=0"^'- "''• (c)m (cO^

(3.21

a I + VI y I < 1-

10

Lauricolla (1893) further generalized the four Appell function Fi, F2,

F3, F4 to function of n variables.

Fx \a,bi,- •• ,br,\ ci, •••,€„; Xi.---.Xr

E (111.11)2- Hl„=i} \C] )nii ' ' ' {(^Vjllln ^ " ^ 1 - >^hT

xi I + - - - + |,x„ | < 1 ; (3.22)

Fjj [ai, • • •, a„, 6], • • •, 6„; c; X], • • •, x„\

^ ( a i ) , » , • • • ( a r 7 ) ; » „ ( ^ ] ) m r - - ( ^ 0 / » „ ^'P < "

;/;],m2. ,«'„=0 iCj„, j+ _l „,^ 77?,]. 777., .

maxjl 3:1 |, I .T2 I, • • •, I x„ 1} < 1; (3.23)

F^\a,b; ci, ••-,€„; XI , - - - , .T„ ]

nii,tii2, , " ' , ,=0 ( C i j m , • • • [Cr,),i,^.^ Vfli. rrir

Xi + \/U„ I < 1 (3.24)

F^j^\a.bj,--- ,6„: c: a: i , - - - ,2:„]

_ ^ ( Q ) W I + + » > „ ( P I ) ; » I • • • {K)iii„ ^l ^ ^77 "

max{| .X] | , - - - J ' / 1} < 1; (3-25;

Clearly, we have

f f = F„ff = f„f ) = F„FS' = Fu

11

F (1) A Fi-P = FP' = f i " = ,F, B C 2-f^]

Also, we have

.(3) F y [ a , 61,62-?>i ;c , x,?/,2^

CXD 0 0 0 0

kl ml nl S 5: E , . (3.26)

Just as the Gaussian 2-F1 function was gencrahzed to ^F^ by increasing

the number of numerator and denominator parameter, the four Appell

functions were unified and generahzed by Kampe de Feriet (1921) who

defined a general hypergeometric function of two variables.

The notation introduced by Kampe de Feriet for his double hypergeo­

metric function of superior order was subsequently abbreviated in 1941 by

Burchnall and Chaundy [9]. We recall here the definition of a more gen­

eral class double hypergeometric function [than the one defined by Kampe

de Feriet] in a slightly modified notation [see for example Srivastava and

Panda [155]

Fi •P Q ^ l,iii,n

a. {bqYich) x,y

ai) • il3,n)]M

00 00 U{aj)r+s n{bj)r n ( c ^ ) s r„ . s

ri si

(3 27)

where, for convergence

i ) p + g < / + r77 + l , p + / c < / + n + l, | j | < o o , | 7 / |<oo , or

12

ii) p + q = I + m. + l^ p + k = 1 + n + 1, and

(3.28) I ^ r""' + I y \^^-'^< 1 , if jD > /

max{| X \,\ y \} < 1 , if p < I

Although the double hypergeometric function defined by (3.27) re­

duces to the Kampe de Feriet function in the spaecial case:

q = k and m = n,

yet it is usually referred to in the Hterature as the Kampe de Feriet

function. Similarly, a general triple hypergeometric series F^''''^[x^y, z] (cf.

Srivastava [152], p.428) is defined as:

a)::{by,{b'y,{b"):{dy,{d')-{d") ; F^'^\x,y.,z] = F^'^^

{e)-,{gy.,{g'y{g"):{hyih'y{h'' x,y,z

CO OO OO rplll yt^ yP

= E E E A(m,n,p)—-y —, (3.29)

where, for convenience

A D n {Q'j)n)+v+p n {bj)ni^ri H [o^j^+p Y\ [bj)p-\-„,

A{m,n,p) = ^ ^^ g^ g^ n+n+p U^{9j)n,+v U^{g'j)ii+p n{gj)p+ni

X

D D' , D" n idj),n u {d'X n (d''p

1=1 ' 1=1 J = l ;-/ D' H"

n [hj)„ n {d!j)r, n {h"^)p ] = l 7 = 1 J = l •

. 7 - 1

(3.30)

13

where (a) abbreviates, the array of a parameters ai. 02, • • ' , Oyj, with sim­

ilar interpretations for (6), (5'), (5"), et. cetra. The triple hyper geometric

series in (3.29) converges absolutely when

1 + E + G + G" + H ~A~B-B"-C > 0 1 + E^G + G' + H'~A~B-B'-C' > 0 1 + E + G' + G" + H"-A-B'~B''- C" > 0

(3.3i;

where the equalities hold true for suitably constrained values of | a: |, \y

and \ z \.

In this thesis we also need to extend the definition of Kampe de Feriet

double hypergeometric function to generalized triple hypergeometric func­

tion. We define the extended form as follows:

F ((a^), a, /3,7) : ( f e ) , <5, e); ((c^), V. 9): {{do)., 0, p) : ((e,,), 0 ; ( ( / F ) , V ); ((/i//),A,/i,z/) : {{ij).X.,T)-,{{lL).,t,iL):{{rnM).,v,w) : ((UA,), a ) : ((p^), A ;

(te),x) ;

(to),V): x,y, z

A B 00 00 00 11 V^jj''i''+''ir+7s

= E E s: ' ' 7=1 j = l

r* iJ E F G Ti{Cj)r,k+0. Yl{d^)^r+psU{ej)^k n (/ j) , / . r ri(.9y)xs ^A „,, . s

j = i T^ 1^ M jf^i X y z M , ^ N , ^ P , ^ Q , ^ k\ r\ s\ L

(3.32) n {1j)fk+v, n {ruj) n (^J)(TA- n {PJ)AI n (<??)vs

j = i j = i j==i j= i 7=1

For Q' = /3 = 7 = ^ = f = ^ = ^ = 0 = /> = < = '( = X = ^ =

^ = iy = ( = T = t = ii = i; = 'UJ = (T = A = V = l, (3.32) reduces to

14

the three variable version of the general form of the Kampe de Feriet's

double hypergeometric function.

4. A STUDY OF A T W O VARIABLES L E G E N D R E POLYNO­

MIALS: The Hermite polynomials H„{x) and the Legendre polynomials

P„{x) are respectively defined by

e- - '= = E ^ ^ (4.1)

and . oo

( l -2 ,T t + t2)"2 = Y.Pr>{x)f (4.2) n={]

A careful inspection of the L.H.S. of (4.1) and (4.2) reveals the fact

that L.H.S. of (4.1) is e ' and that of (4.2) is (1 — w)'2 where u = 2xt —

t^. Thus Hr,{x) and P„{x) are examples of polynomials generated by

a function of the form G{2xt — t^)- The expansions of (1 — i/)~2 and

1 ~ u — vY~^ are given by

~ il] if (1 -^n = E^^^Y- (4.3)

77=0 ^ '

and

£S Si [2>„+t '" '' 1-n-V)-'- = s: E -^^^hr- f^-^)

T7=0/r=0 "- ' ^ •

The expansion (4.4) motivates a two variable analogue of Legendre

polynomials by taking u = 2xs - s and v = 2yf - f in (4.4). In Chapter

15

II an attempt has been made to define two variable analogues of poly-

nomaials by means of generating functions of the form G{u,v) where

u = 2x8 — s^ and v = 2yt — t^ and then a particular example of it namely

the two variable analogue of Legendre polynomial has been considered.

Thus chapter II deals with a study of a two variable polynomial

P-n,k{^-y) analogous to the Legendre polynomial P„{x). This chapter

contains differential recurrence relations, a partial difTerential equation,

double generating functions, double and triple hypergeometric forms, a

special property and a bilinear double generating function for the newly

defined polynomial Pr,_k[x.y).

5. A NOTE ON A CALCULUS FOR THE rfc,:^,y-OPERATOR:

In 1964, VV.A. Al-Salam [2] defined and studied the properties and appli­

cations of the operator

e = x(l + xD), D = ~ (5-1) ax

He used this operator very elegantly to derive and generalize some

known formulae involving some of the classical orthogonal polynomials.

He also gave a number of new results and obtained opeartional represen­

tations of the Laguerre, Jacobi, Legendre and other polynomials.

In 1971, H.B. Mittal [121] generalized the operator 0 by means of the

16

relation

n = x{k + xD) (5.2)

where A: is a constant. He used this operator to obtain results for gener­

alized Laguerre, Jacobi and other polynomials.

In 1992, M.A. Khan [50], the first author of this paper, introduced

g-extension of the operator (1.2) by means of the following relation:

Tk,,.r = x{l-q){\k]-Vq'xD,,}, (5.3)

where k is constant, | g |< 1, [/c] is a g-number and Dq^r is a g-derivative

with respect to x. Letting (7 — 1, (1.3) reduces to (1.2).

The paper [50] is a study of a calculus for the T/rg^T-opeartor. Later,

in a series of papers M.A. Khan [55],[60],[69] used this operator to ob­

tain operational generating formulae for g-Laguerre, g-Bessel, g-Jacobi

and other g-polynomials. In particular, various generating functions and

recurrence relations were obtained for g-Laguerre polynomials.

Chapter deals with a calculus for a two variable analogue of the oper­

ator (5.2). The aim is to use this operator and the results obtained here in

various opeartional representations, generating functions and recurrence

relations of the two variable polynomials as a part of further study in this

direction.

17

6. ON A NEW CLASS OF POLYNOMIAL SET SUGGESTED

BY THE POLYNOMIAL OF M. A. KHAN AND A.R. KHAN:

Recently in 2002, M. A. Khan and A. R. Khan [93] studied a new class

of polynomials A^^{x) suggested by Gegenbauer polynomials C,' (.T) and

defined by means of the generating relation

CxD

1 - 3xH + Sxt'- r)"" = Y^Ki^W (6-1) 77 = 0

This motivates us to further generalize A':^{x) and the chapter IV is

an outcome of such a study of another new class of polynomials M^(.x)

suggested by the polynomials A, (x) and defined by means of generating

function of the form G{4:X^t — Qx'^t'^ + ixt'^ — t'^) for the choice G{u) =

l-ii)~r

Further, one needs the following results:

OO OO 0 0 (ry\ J.J n,^ yf

1 - :r - y - ^)-" = E E E ^ ^ ^ ^ ± 7 ^ ^ ^ ^ (6.2) r=0.s=0 j=0 r\ SI ]\

For the special choice a = ~n where n is a positive integer (6.2)

becomes

OO n—r n~r—s ( n^\ , o-J nr'' y'^

1^ 1^ L^ r-l QI T1

r=0.s-=0 ji=0 r. S. J.

l-x-y-zf = EE Y. ' " ^ t ; ^ !i ^' ' (6'3)

18

The following theorem given in the paper of M. A. Khan and A. R.

Khan [93] is also needed

THEOREM: From

oo

it follows that g'^ix) = 0, xg\{x) = 3gi{x), x^g'^ix) = ^xg^ix) + 6(7](x)

and for n > 3

x^%{x) — 3na:^5'„(a:)

= Zx^g^^^ix) - 3(n - l)xg^^i{x) - Zxg„^^{x) + {n- 2)g„^2{x) + g'^^i^)

(6.4)

Chapter IV deals with a study of a new class of polynomials M^^{x)

suggested by the polynomial A'^{x) of M.A. Khan and A.R. Khan and

defined by means of a generating function of the form G{4x'^t — 6xH'^ +

ixt"^ — t'*) for the choice G{IL) = (1 — u)^'^. The chapter contains some

interesting results in the form of recurrance relations, generating function,

triple hypergeometric forms, integral representations, Schlafli's Contour

Integral and Laplace transforms.

7. ON A GENERAL CLASS OF POLYNOMIALS OF THREE

VARIABLES 2:( "' '' '' ' ' )(x, y, z): In 1972, T.R. Prabhakar and Suman

Rekha [133] considered a general class of polynomials suggested by La-

19

guerre polynomials as defined below

,jt ^ Tjan + 0+1) " {-v),.T^

^" *•"' ~ S h,k}r[ok + 0 + i) (^-i*

and proved certain results for these polynomials. Later, in 1978, they

[134] derived an integral representation, a finite sum property and a se­

ries relation for these polynomials. Evidently, if cc = /c, a positive inte­

ger, then U^'^\x^') = Z^^{x] k) which is Konhauser's polynomial and more

particularly VJ\x) — L\['^\x) where L\[^\X) is the generalized Laguerre

polynomials.

In 1991, S.F. Ragab [135] defined Laguerre polynomials of two vari­

ables L^"' ^ (x,?/) as follows:

4».«(,,,) = E(!i±^+W"- + /' + i

y- {~yj Lr,-r{x)

h r\ V{(y + 71 - r -f-1) r(/? + r -M) ^ ' ^

where L^"''(.x') is the well known Laguerre polynomial of one variable.

It may be noted that the definition (1.2) for L^^^^-'^\x^y) is equivalent

to

where n is any positive integer or zero.

20

In 1997, M.A. Khan and A.K. Shukla [73] defined Laguerre polyno­

mials of three variables Ll^^'^^'^\x, y, z) as follows:

r(«,/i,7)r^ „ ,\ - («±11"(/^ + 1)"(7 + 1). iv„ {X^ //, Z) —

77 n—rv—r—s

\\i

r=oto to r\ s\ k\{a + 1),(/^ + 1),(7 + 1). ^ ' ^

They established various generating functions and integrals represen­

tations for these polynomials. In 1998, they further generalized these

polynomials to m-variables [74].

Recently, in 2002, M.A. Khan and A.R. Khan [100] defined and stud­

ied three variables analogue of Konhauser's polynomial as defined below:

z;-«-(.,,..;A:,p,,) = r(fa + a+j)r(;.» + ff + i)r(,n + , + i: ,n! \3

77 n~r n—r—s

,=0tt, ,^1 rh\jW{kj + a+ l)ri:ps + 0+ l)r(qr + 7 + 1) ^ ' '

which for fc = p = g = 1 reduces to (1.4)

Motivated by the above investigations the chapter V is a study of

a general class of polynomials of three variables L^^^'^^'^-^'^-^^\x,y,z) sug­

gested by (7.1), (7.2) and (7.4). The polyuomial 4^-/^'^'^'^''^)(.T, y, z) yields

in particular cases the above mentioned polynomials (7.1),(7.2),(7.4) and

(7.5).

Thus chapter V deals with a study of a general class of polynomial

li»AiAXi^)(^x,y,z) of three variables which include as particular cases

21

various polynomials of one, two and three variables studied earlier by a

number of mathematicians. Prominent among them are the polynomi­

als L'^'^\x) due to Prabhakar and Rekha, L^','^^\x, y) due to S.F. Ragab,

Zli{x]k) due to Konhauser, Ll]{x) due to Laguerre. Z'^-f^''^{x,y^ z:k,'p,q)

due to M.A. Khan and A.R. Khan and three and m-variables analogous of

Laguerre polynomials due to M.A. Khan and A.K. Shukla. The chapter

contains generating functions, a finite sum property, integral representa­

tion, Schlafli's contour integral, fractional integrals, Laplace transforms

and a series relation for this general class of polynomial.

8. SOLUTION OF MISCIBLE FLUID FLOW THROUGH

POROUS MEDIA IN TERMS OF HYPERGEOMETRIC

FUNCTION iFii In chapter VI a specific problem of miscible fluid

flow through homogeneous porous media with certain assumptions have

been discussed. The mathematical formulation of the miscible fluid flow

through porous media gives a non-linear partial difi^erential equation,

which is transformed into ordinary differential equation and reduced to

a particular case of generalized hypergeometric equation (sec Rainville

[141]) by changing the independent variables using subsequent transfor­

mation. Finally, one get a general solution in terms of hypergeometric

functions o^i is obtained.

22

The hypergeometric dispersion in the macroscopic outcome of the

actual moment of individual tracer particle through the pores and various

physical and chemical phenomenon that takes place within the pores.

This phenomenon simultaneously occurs due to molecular diffusion and

convection. This phenomenon plays an important role in the sea water

intrusion into reservoirs at river mouth and in the underground recharge

of waste water discussed by Scheidagger [148]. Here the phenomenon

of longitudinal dispersion which is processed by which miscible fluid in

laminar flow mix in the direction of flow.

Several authors have discussed this problem in different view point,

namely Scheidagger [148] and Bhathawala [7].

23

CHAPTER II

A STUDY OF A TWO VARIABLES

LEGENDRE POLYNOMIALS

ABSTRACT: The present Chapter deals with a study of a two vari­

able pol,ynomial P„i-(.T,7/) analogous to the Legendre polynomial P„{x).

The chapter contains differential recurrence relations, a partial differential

equation, double generating functions, double and triple hypergeometric

forms, a special property and a bihnear double generating function for

the newly defined polynomial Pj,^i.{x,y).

1. INTRODUCTION: The Hermite polynomials Hr,{x) and the Leg­

endre polynomials P„(x) are respectively defined by

0 0 2Tf-f2 _ y - Pr){X)t ,

and CX)

l-2xt + f/)--^ = Y.Pv{x)f (1.2)

A careful inspection of the L.H.S. of (1.1) and (1.2) reveals the fact

that of L.H.S. of (1.1) is e^' and that of (1.2) is (1 - n)^2 where u =

2xt - i^. Thus Hr^(x) and Pr,[x) are examples of polynomials generated

by a function of the form G[2xt - f). The expansions of (1 - u)~~^ and

[l — u — v) 2 are given by

1 (X) ll) y "

ii-uy^ = E^^^ (1.3) =0 " •

and

i-u-vY^ = TY.^-^^^,— (1.4)

The expansion (1.4) motivates a two variable analogue of Legendrc

polynomials by taking ii = 2x5 — s and v — 2yt — t^ in (1.4). Thus we

first attempt to define two variable analogues of polynomaials by means

of generating functions of the form G{IJ,V) where v = 2xs — s and

V = 2yt — t^ before embarking on a particular example of it namely the

two variable analogue of Legendre polynomial.

2. DOUBLE GENERATING FUNCTIONS OF THE FORM

G{2xs — s^, 2yt — t^)\ Consider the double generating relation

oo oo

G{2x, - s\ 2yt -i') = E X : 9ni C , 'jVi' (2 1) ?7=0A=0

in which G{u, v) has a formal double power series expansion where a

formal double power series in one for which the radius of convergence is

not necessarily greater than zero. Thus G determines the coefficient set

{gn,k{x, y)} even if the double series is divergent for s 7 0, t 7 0. Let

25

F = G{u,v), where

u = 2xs — s^ and

V = 2yt-t^

>

Now, from partial differentiation, we have

dF dFdu dFdv dx du dx dv dx dF dFdu dFdv ds du ds dv ds dF dF du dF dv dy du dy dv dy

dF dFdu dFdv dt dv dt dv dt

(2.2:

(2.3)

(2.4)

(2.5)

(2.6)

substituting the values of partial derivatives of u and v the above equa­

tions reduce to

(2.7)

(2.8)

(2.9)

(2.10) 01 uv

Multiplying (2.7) by {x ~ s) and (2.8) by s and subtracting, we get

0 (2.11

dF dx

dF ds dF

dy dF

^4-

= 2s ^^ du

. dF

dF dv

^ dF = 2(2/ t) ^ ,

dF dF x — s] ~ 5 dx ds

26

Similarly from (2.9) and (2.10), we get

, dF dF ( ^ - * ) a ^ - * a r = «' (2.12;

Since 00 00

F = G{2xs-s\2yt-t') = E E 9.Ax,y)s"t n={)k=l}

it follows from (2.11) that

CXD ^ 0 0 / ^ CX3

or

cxo / ^ 0 0 0 0 0 0 / ^

(2.13)

Similarly, from (2.12), we obtain

00 / ^ 00 00 00 ^

E y^Pr^A^.yVi'- E A:P,,,(x,y)5¥^=E E:^i^..A^-i(^,i/)5"t^' r(,/r=0 "^l/ n,k=Q n-^i] k=l ^V

(2.14)

Equating the coefficients of s^t''' in (2.13) and (2.14) we obtain the

following results:

T H E O R E M 1. From

(X) 0 0

G{2xs-s\2yt-t'] = EEg^A^^yV^' 77 = 0 A" = 0

27

d d it follows that ~—g()^i,{x,y) = 0, k > 0, —-gr,fi{x,y) = 0, n > 0 and for

??, A: > 1 ,

^'r^gr,,k[x,y)~ng„,k[x,y) = -^gr,-i,k[x,y) (2.15)

and

d d yY%.k{^,y)-^9vA^^y) = o-^n,A-](3:,?/) (2.16)

Adding (2.15) and (2.16), we obtain

( d d\ d d {x-r^ + y-r^ j 9v.k(x,y)-{n+k)g„ i,(x,y) = -—g„^ij,{x,y)+--g„,i,-i{x.y)

(2.17)

The differential recurrence relations (2.15),(2.16) and (2.17) are com­

mon to all sets g„^kix^ y) possessing a generating function of the form used

in (2,1). In this chapter we shah consider the polynomials gr,k{^,y) for

the choice G{u, v) — {\ — u — v]

3. THE LEGENDRE POLYNOMIALS OF TWO VARIABLES:

We define the Legendre polynomials of two variables, denoted by Pu,k{x. y),

by the double generating relation

OO 0 0

(1 - 2xs + s^- 2yt + ^2)-o = J:E Pruk{x^ y)s''t' (3.1)

in which {l-2xs+s'^-2yt+t'^)~''~ denotes the particular branch which -^ 1

28

as 5 -^ 0 and t ^ 0. We shall first show that P„_k{x,y) is a polynomial

of degree precisely n in x and k in y.

bmce (1 - z/, - ?;) " = E E r n . we may write

11=0 k=o nl kl

^ y y y y ( 1 ^ 2 x 5 ) " (2^^)'^(-n), (-k), fsYfty -

r/=0A-=0r=0.7=0 n! A:! r! j ! \2x/ \2yJ

,,=() A=0 r=0 7=0 f J K"' - ^) • (^ - J) •'

oo 00 i2i i2i (7,) ,, {2xY-^'{2yY-'^^{-lY+^s'' t'' 11] \t] fV 2/n+k—r—j

SSS.S r\ j\ (n ^ 2r)\ {k - 2j)\

We thus obtain

r=o,7=o r] j \ [n - 2r)\ [k - 2])]

from which it foUows that P„^k{x, y) is a polynomial in two variables x and

y of degree precisely n in x and A; in y. Thus P„,k{x,y) is a polynomial

in two variables x and ^ of degree n + /c. Equation (3.2) also yields

P , , (x , , ) = ^ ^ S ^ - ^ + - ' (3.3) n! k\

29

where TT is a polynomial in two variables x and y of degree ?? + A- — 2.

If in (3.1), we replace x by ~x and s by —5, the left member does not

change. Hence

P.A~^.y) = {-\TP,k{x,y) (3.4)

Similarly by replacing y by —y and t by ~t in (3.1), we obtain

P.A^;-y) = i-ifp.A^-.y) (3-5)

so that P„/,.(a:, y) is an odd function of x for n odd. an even function of

x for n even. Similarly, P„^/^(x,i/) is an odd function of y for /c odd, an

even function of y for k even.

Similarly replacing x by —x, y by —j/, .s by —s and t by —/ in (3.1),

we obtain

P.A-^^-y) = Hr^'P.A^,y) (3.6)

Puttmg / = 0 in (3.1), we get

p„,oi^,y) = PA^) (3-7)

where P„{x) is the well known Legendre polynomial. Similarly by putting

5 = 0 in (3.1), we get

Po,k{^.y) = Pk{y) (3.

30

From (3.1) with x = 0 and y = 0, we get

l + s2+ t2wi 77 J.A

r/=0/,=0

But (1 + 5 + / 2 I ^2^ oo oo ( - 1 ) " + ^ f l ) , s 2 . p r

77=0 fr=0

2Jri+k

n\ k\

Hence 277 + 1,2^(0,0) = 0 , P2«2A+l(0,0) = 0 , P2774],2/,-+l(0,0) = 0, 1

^ ^ " " ^ (2/77+^-^2.,2A^(0,0) =

Equation (3.2) yields

(3.10)

dx

[^1 [|] (-1)'^+? (i Pvk[x,y) = J2 H

2)n+k—r—j 2(2xf~^'^'{2yf~^3

r=0 j=() r\j\ ( n - l - 2 r ) ! (fc - 2,;) (3.11

dy

[fl [¥] i^r7.(:r.^) = E E

^iy+] (i 2/v+k—r'

'n-2rn(n \k-l-2j ^ {2xr-''%2y)

r=() j=0 r! j \ (?7 - 2 r ) ! (A:- 1 - 2j)! (3.12;

d ox

^_iy>+k2fi 2Jn+k+l __

r=0,(/=0 n ! A;!

dx P2v,2k{x,y)

1 = 0 y=()

d ^P2v.2k+l{x,y) ox

0,

0, 7 = 0 7y=0

(9 ^ i^2 r7 +1,2AM-1(3^,?/) = 0.,

7 = 0 (/=()

-1\77+A' fr ^ l2y„+A'

n! k\

(3.13)

31

Similarly,

d

d

d

-^P27i+i,2k+]{x,y) and

r=0.y=0

0,

0,

0

^P2n.2k+l{x,y) r={),y={} n! k\

•I)""' (i;„,, n! k\

(3.14)

4. DIFFERENTIAL RECURRENCE RELATIONS: From Theo­

rem 1, it is evident that the generating relation

oo oo n^k l--2xs + s'-2yt + t'y-^ = EEPnA^.y)s''t

77=0 ^-=0 (4.i:

impHes the differential recurrence relations

d d X TT- Pn.k(x, y) - nP„,k{x, y) = ~- P„^i^k{x, y). dx dx

(4.2)

d d y TT PvA^^ y) - kPr,.k{x, T/) = — Pr,.k^) (x, y),

oy oy (4.3)

and

^ d d \ ^-^ + y ir] ^"•' ( ' y)-i'^ + k)P^,kix, y)

V ox oy/ 32

d d = ^ ^"-i./.-(3^,?/) + gy Pn,L^i{x.,y) (4.4)

From (4.1) it follows by the usual method (differentiation) that

g OO (X) P

1 - 2xs + 2 - 2yl + t')'-^ = EEQ^ P..k{x, y)s---h' (4.5) n={) k--=0

oo oo (^

n=0 /r=() ^ ? / (1 - 2.T5 + 5 - 27/ + ^2^-5 = E E TT Pn.ki^-, y)s"t''' (4-6)

oo oo

{x -s){l- 2xs + 5' - 2yt + i')-2 = E E ^ r,.A (. , ?/)«"'" i (4.7)

(2/-i)(l-2:r.s + 5^-2?yt + t 0 - ^ - E Y. kFr.A^^y)'^"P (4-8)

Since l-s^ -t^ ~ 2s(x ~ s) ~ 2t{y -t) = l - 2sx + s^ - 27/t + f, we

may multiply the left member of (4.5) by 1 — 5 , the left member of (4.6)

by —] , the left member of (4.7) by —25, the left member of (4.8) by —2f

and add and obtain the left member of (4.1). In this wav we find that

00 /^ CO f) oo ^

E |-^,a(^:i/)5"-V^-E j^P.A^.,y)s''-'h'-Z ~ p„A^,y)s^t''-' rKk={) OX „/,.^0 (^^ v,k={] Oy

oo OG oo

~ E 2nP,Ax,y)s'H'- E 2kP,,jXx,y)s''t'= E PnA^^vV^' n,k={) n,k={} v.k=0

or

oo f) oo f) oo f)

E ^P.A^,y)s''~h'-j: P,A^.^y)s^+h'-j: -p„,,(.T,y)/'t'''^ n.k^i) OX n.k={) OX „,k=i) Oy

= E {2n + 2k + l)P,.k{^,y)s''t

33

We thus obtain another differential recurrence relation

d d d

(4.9)

Similarly, we can get

O d d {2n + 2k^l)P„ i,(x., y) = -~Pr,^k+\{x. y) - ~-P„^i^k(x, y) - ~Pr,j,-i{x, y)

oy ox oy

(4.10)

Adding (4.9) successively to (4.2).(4.3) and (4.4), we get

d d d x-T-Pr,.k{x,y) ^ —P„+i,kix,y) - ~Pr,^k^\{x,y) - {n + 2k+ l)P„k{x,y)

(4.11) d d d

y~^Pn.k{x, y) = —P„+]i^(2;, y) - ~Pr,^i.k[x, y)-[2n + k + l)Pn.k(x, y)

(4.12)

U — + ?/ ^ 1 Pn^k(x.y) = — P„+u(x, t / )-(?i + /c + l)P„i(.T,7/) (4.13)

Adding (4.10) successively to (4.2),(4.3) and (4.4), we obtain

d d d x~-P,,j,{x,y) = --P„j,+i{x,y) - -^P„^k-]ix,y) - {n + 2k + l)P„^k{x.,y)

(4.14)

d d d y~-P^^k{x,y) = —Pn.k+i{x.,y) - ~P„^i,k{x,y) - (2n + fc + l)P„,^(x,?/)

(4.15)

[^-^ + y-^) PnA^^y) = ^ P.,^+](^,?/)-(n + /c + l)P,,,(.x,y) (4.16)

34

Shifting the index from n to n — 1 in (4.11) and using (4.2), we get

d d X -l)^Pr,A^,y) -=nxPr,^k{x,y)'-^Pr>-],k^i{x,y)-{n+2k)P^^i^k{x,y)

dx dy' (4.17)

Similarly shifting the index from /c to A; — 1 in (4.15) and using (4.3).

we get

d d (y -l)~Pn,k[x,y) = kyP„,k{x,y)-~P„-i,i:^i{x,y)~{2n+k)P„,k-i{x,y)

dy dx (4.18)

Adding (4.17) and (4.18), we obtain

x' - i)Tr + iy' ~ ^)^] P-A^^y dx dyl

( d d ] nx + ky)P^,k(x,y) - l~ + — )Pr,~\.k-\(x.,y)

-{2n + k)P„^k~\{x,y)

n + 2fc)P„_i.A.(.T,t/)

(4.19)

5. PARTIAL DIFFERENTIAL EQUATION OF Pn,k{x, y): From

(4.2) and (4.3), we have

d d — P„^i,fr(x, y) = x— P„A^., y) - nPr,.k{x, y) dx dx

d^ d^ d j^.P.-iA^.y) = x^^Pr,^k{x,y)^{\-n)~Pr^^k(x,y)

d d — P^j,^i{x,y) = yTT Pr^.k{^^y)-kP„j,{x,y) dy dy

^,P.,-ii^.y) = y^,PnA^-^y) + i^~k)l^P^,{^.,y)

(5.i;

(5.2)

35

Shifting the index from r?. to n — 1 in (4.11) and from A; to /c — 1 in

(4.15), we get

d d d x--P„_ i^k{x, y) = ^-P„,/,(.T, y) - (n + 2/c)P„_i k{x, y) - --F„_ii^„] (x, y)

ox ox oy (5.3)

d d d y^Pr,,k^iix,y) = —P„,^.(T,I/) - (2n + A;)P„,A._i(,T,7/) - -^P„_i^k-i{x.,y)

oy oy ox (5.4)

Differentiating (5.3) partially with respect to x and (5.4) with respect

to y, we get

.X—^P„_],A-(.x, ) = —^P„,k{x,y) - (n + 2/c + 1)—P„_i,^(a:,y)

Pn-].k-i{x,y) (5.5) dxdy

Q2 Q2 Q y-Q^Pv,k-i{x., y) = -^Pr,.k{x, y) - {2n + A; + l)^P„,;;.„i(2:, y)

Pn^\,k^i[x,y) (5.6) dxdy

Using (5.1) in (5.5) and (5.2) in (5.6), wc obtain

(1 - x')^^PnAx, y) - 2(1 + k)x^Pr,,k{x., y) + n{n + 2k + l)P,,k{x^ y)

^' P.^LkA^.y) (5-7) dxdy

(1 - y^)^,P..k[^., y) - 2(1 + n)y^Pn k{x,y) + k(2n + k + l)P„,,(x, y) dy^ oy

^' P.-i,k^i{x,y) (5-8: dxdy

36

Subtracting (5.8) from (5.7), we obtain

• 2 ( 1 + k)x-- - (1 + n)2/— P„,/,(i:, i/) + (n-A;-)(7i+/c+l)P„,,(,T, y) =-0

(5.9)

Here (5.9) is the partial difTerential equation satisfied by P„j{x,y).

6. ADDITIONAL DOUBLE GENERATING FUNCTIONS: The

generating function {l~2xs + s'^-2yt + t'^)~2 used to define a polynomial

Pn^ki^iV) in two variables x and y analogous to Legendre polynomials

Pr,{x) in a single variable x can be expanded in powers of 5 and t in new

ways, thus yielding additional results. For instance

0 0 CXD

E E Pr,,k{^,y)s''t' = {l-2xs + s'~2yt + t')-^-

1-xs- ytf ~ s^{x'^ - 1) - t ^ ( / - 1) - 2xyst

s'^ix^ - 1) f^iy'^ ~ 1) 2xyst {l—xs~yt) - 1

1 — xs — yty (1 — xs — yty (1 — X5 — ytf-

- - - (1),+,+. ' ( ' - m'' {y" - 1) ^ (2 ;i/ '0' ,=0/t^).t^) j ! P! ^!(1 - ^8 - ,yt)2?+2,+2r+1

E E E E E -'^^^ oo oo 00 oo 00 | i ) . , (l + 2j + 2p + 2r)„+A^s"+2j+'^(:c2_;Ly

Ti=oA-=oj=Oi;=Or=o ,7' p! ^! ^! ^!

37

s '

= E ~ (i)j+p+r (l)"+A^+2,+2p+2r5"+2i+'^(^2 _ l ) . ^ U 2 ; . + . ( ^ 2 __ iy2r^^n+ryk+r

,.p.rvk=[) ]\p\r\ n\ k\ (l)2;+2p+2r

Equating the coefficients of s"t^'', we obtain

(6.1)

Let us now employ (6.1) to discover new double generating functions

for Pjjj{x,y). Consider, for arbitrary r, the double sum

„ OA-=o (n + /c)! v={}k=i) {n + ky.

^ - - - - - (c)„+/.+2,+2/.+2r(^^ - l)^(t/^ - l)y.^»+r^^-+r^n+2,+r^fr+2,+r

h h h h h 2^J^^P+^{l),^,,_rj\ p\ r] n\ k\

^ff^f {c)2jy2p+2r{x^ - iy{y' - iyf^yYs^-^^H''>'-

^ ^ {c + 2j + 2p + 2ry-,+k(xsY{ytf X E E

71=0 k=i) ri\k\

38

oo oo oo

22,+2,+.(i) _^^^^^ -, , ,

cx) oo oo ( I 1

= (1 - xs ^ yt)- T.EE ^'',;r ' -I %'T

X i'(y^ - 1 2.T2/5t

1 — xs — yt)^ j [ (1 — xs — j/t)2

52(^2 - r

! 1(1 — ^5 - ?/i)'

= {l-xs-yty

X F(3)

c c 1

2 ' 2 ^ 2 ( l - - r s -7y / )2 ' ( ] - r s ^ y ) 2 ' ( l ^ r ' , - ^ / ) ^

where F ' -'fx,^, z] is a triple hypergeometric series [cf. Srivastava [158],

p.428j. We have thus discovered the family of double generating functions

(1 — xs — yt) '

c c 1 2 ' 2 ^ 2

X JPC^) — ) — 1 — )

S - ( T ^ - I ) 1-iv--l) 2iv',f {l-rs-yty ( ] - r s - 7 / / ) 2 ' ( l--7s-7//)2

1

n={]k={) in + k). (6.2

which c may be any complex number. If c is unity, (6.2) degenerates into

the generating relation used to define P„,j,{x,y).

39

Let us now return to (6.1) and consider the double sum

j+p+r

X

jV-2] k-2pgr)jk

]\ p\ r\ [n ~ 2'j — r)\ {k — 2p ~ r)!

OO CX) OC OO _ 2^ (a:^ - l ) ^ ( t / 2 - iy^r,+ryk+rgn+2j+r^k+2p+r

~ h hhh h 22^^2^^)^ _^ -, p, , , ! .,

~ ^ ^ ( x ^ - iyfa^-iy(^^g^)''g^^^^^' v2 ^ (2:5)"(?/t)^

y=0p=0r=0 22;+2/j+rQ

_ Ts+yt 0 0 CX) OO

gT.s+vf y^ y- y^ 1 2f'2 ^\^J (+2f,,2 1 ^ U '

^ ) p = ( ) r = 0 j ! p ! ' ' ^ ' Wj+P +r

r(j/^-i) r ri;?/5tr' 4 ' M J

= e T-'^+yi pi'i) s (x^ ~ 1) t''(z/'' — 1) xyst

(6.3)

7. TRIPLE HYPERGEOMETRIC FORMS OF Pu^k{x,y): We

return once more to the original definition of Pn^kix^y)'.

J OO OO

l-2xs + s'~2yt + t^)-'i = ^ j;p„,,{,T,i/)s"t'' (7.i: 71 = 0/, =0

This time we note that

l~2xs+s^-2yt+t'^ 1 - s - t)2 - 2s(a: - 1) - 2t{y - 1) - 2st

40

[i-s-tr' 1 -2s{x-l) 2 / ( z / - i ; 2sf

( l - 5 - t ) 2 ( l - 5 - i ) 2 ( 1 - 5 - / ) ^

which permits us to write

n=0 fr=0 j=0/)=Or=0 j \ p] r\ (1 ~ S - i)2?+2/J+2r+l

oo cx) oo ex. cx) 2""" ' (D,+„+. i^ ~ 1)' {y - 1)" 71=0 k=(] j = 0 p=0 r =0 .7 •' ' ^•

X 2j+2/j+2r n ! A;!

rij-k

„={)k=Oj=Op=o r=() j ! p! r! (n ~ J - r)! {k - p - r)! (l)2j+2/;+2r

„fo At ) jt^);S] r'S) ^21+1'+^ (l),+pH_, j ! p! r! (n-j- r)! ( ^ - p - r)!

Therefore

P„ A (a:, y) =

X ir(^)

n +A-)!

n\ k\

:: —n; —fc; 1 + n + A: : _; _; 1 — X 1 — ^ 1

~~2~ ' 2 ' 2 (7.2)

Since P„,/.(-:^, -y) = {'If^^ Pv /. C - v)- it follows from (7.2) that also

41

PvA^^y)

X i?(3) —n: -/c; 1 + n + /c

1 + x 1 + ?/ 1

2 ''^y~'2 (7.3)

Next, consider (3.2) again

Pr,.k{x,y) = E E SiS(-ir^Q)„H.,-.-,(2^r"^(2y)^-2,

r!,;! ( n - 2 r ) ! (A:-2j)!

We may write it as

r=oj=o r! j ! n ! /c! ( | - ? - fc)^^^

k [?1 (11 /" " 2"""(^)„,,.^"?/ - -

n! /c! EE 2Jr V " 2 "*" 2 J r V 2 J j V 2 "^ 2 7 ^

r=o^=o rljl (l~n- k)^^_^^ x^'^y^-^

or in terms of Kampe de Feriet's double hypergeometric function, we liave

PvA^^y n\ k\

X F

n n 1 k k 1

" 2 ' ~ 2 ^ 2 ' " 2 ' ~ 2 ^ 2 ' 1 1

— n — k L 2

(7.4)

42

8. A SPECIAL P R O P E R T Y OF Pr,,k{x,y): Wc now return to tho

original definition of P„^kix,y) and for convenience use p = (1 — 2xs +

5 - 2yt + fy-. We know that

n=() k=[)

v.k ^ p^l fs.r

X — 5 V — t U V In (8.1), we replace x by , y by , s by - and t by - to get

P ' P P ' P

tZP.J'^^,"—^]p~'-''^"v' v=0k=O P P

2{x-s)ii li^ 2(y - t)u V'

= PIP^ ~2[X- S)U + U^ '-2{y~ i)v + v^]

P' P^ P' +

P'

We may now write

CO OJ

n-OA=() ^ P P '

= [l~ 2xs + 5' - 2yt + r - 2xu + 2us -f u' - 2yv + 2?;/ + v' 2 l ~ i

1 - 2x{s + u) + (s + u)2 - 2?/(i + v) + {t + v 2 i - i

which by (8.1) yields

0 0 CJO

/7 = 0 k={]

^ S y i\ „_-;;-.-i^ „ /,

/? /? n^/ = E En,/,(^/^)(5+^r(^+^

(n + r)! (/r+^-)! P,^rk+A^.y)s' P i^ v^ 0 0 CX3 CXD 0 0

E E E E 77 = 0 /,- = () 7=0 J 0

r! j ! n\ k\

43

p

Equating the coefficients of z/"?/" in the above, we find that

X — s y — t\ -77 —/r—1 r>

P P )

in which p = (1 — 2.TS + s — 2yi + t^)2

9. M O R E G E N E R A T I N G F U N C T I O N S : As an example of the use

of equation (8.2), we shall apply (8.2) to the generating relation

^+V'^(3) 4 ' 4 ' 2

1 - ) • ) 7 )

X — S U — t —SZL

In (9 1), we replace x by . y by , 5 by —^~ and t by

(9.1

P P P and multiply each memeber by p •* where p = {I - 2xs + s^ - 2yt + i

we obtain

P 2\h

p ^ exp

X i?(3)

su(x -

•• —J~

- s) + fv(y -

P'

— • — ) — ) —

-i)

-

9 9

X

4/j^ ' 4/j4 ' 2 / /

00 00

EE n=0 /r=0

r(4A ^—n — k—\ i)"^V p. w ^

/; ' l> 7 —s y—/ \ C"T,"-/-' ' 'I!^' s"w"r?/

n + /c)!

44

oo oo oc c»

„=0^=()r=Oj=0 "'•' ^! ^' j ! (^' + k)\

,7=0 A-=Or=o j=o ("' - r)\ {k - j)\ r\ j] (n^k-r-j)]

^ ^ " ^ (-1)' +' n! A;! Pn,kix,y) vr v'' 5" t^^

^ ^ " ^ ( - n ) , (-/c)jP„.^.(3:,^)u^t;^5"t^-

cx) 00

E E$2[-n,-A;: 1; u,v]P,,,ix,y)s" t' r)=0/r-0

where $2 is one of the seven confluent forms of the four Appell series

defined by Humbert (see [159], pp.25).

This gives a bihnear double generating function.

45

CHAPTER III

A NOTE ON A CALCULUS FOR THE

Tfc,:r,y-OPERATOR

ABSTRACT: The present chapter deals with the study of calculus of

Tk,r,y operator. The operator, which is a two-variale analogue of the oper­

ators given earlier by W.A. Al-Salam [2] and H.B. Mittal [121], has been

used to obtain operational representations for various polynomials of two

variables which are very useful in finding the generating functions and

the recurrence relations of the two variable polynomials,

1. INTRODUCTION: In 1964, WA. Al-Salam [2] defined and studied

the properties and apphcations of the operator

e = x(l + xD), D = — (1.1) ax

He used this operator very elegantly to derive and generalize some

known formulae involving some of the classical orthogonal polynomials.

He also gave a number of new results and obtained opeartional repiesen-

tations of the Laguerre, Jacobi, Legendre and other polynomials.

In 1971, H.B. Mittal [121] generalized the operator 6 by means of the

relation

n = x{k + xD) (1.2)

where A; is a constant. He used this operator to obtain results for gener-

ahzed Laguerre, Jacobi and other polynomials.

In 1992, M.A. Khan [50], the first author of this paper, introduced

g-extension of the operator (1.2) by means of the following relation:

n.,,- = T,il-q){[k]+q'xD,,-}, (1.3)

where k is constant, | g |< 1, [A:] is a g-number and D^^ is a g-derivative

with respect to x. Letting q -^ 1, (1.3) reduces to (1.2).

The paper [50] is a study of a calculus for the T).^^.,r-opeartor. Later,

in a series of papers M.A. Khan [55],[60],[69] used this operator to ob­

tain operational generating formulae for g-Laguerre, g-Bessel, g-Jacobi

and other g-polynomials. In particular, various generating functions and

recurrence relations were obtained for g-Laguerre polynomials.

The present chapter deals with a calculus for a two variable analogue

of the operator (1.2). The aim is to use this operator and the results

of this chapter to obtain various opeartional representations, generating

functions and recurrence relations of the two variable polynomials in other

separate communications.

2. THE rfc,^,y-OPERATOR A N D ITS PROPERTIES: We define

the T/i. r,i7-Operator by means of the following relation:

I d d \

It is easy to verify that

T^ ,, J:c"+'^/+'^} = 2" / ' i ± « l ± ^ ± I j t i ' j ^a^r+Syli+.+. J2.2) V

where r and s are integers, n a positive integer and a and (5 are arbitrary.

It can easily be seen induction that

T^,,, = ^ V n V + ' + + 2j-), ^^ .^^ , ^' = y^ (2.3)

Let F{x) be a function wliich has a Taylor's series expansion, then we

have the formal shift rules:

F{T,_r,yWy^'f{x., y)] = x^^/F[T,,,, + (« + ^)xy]f{x, y) (2.4)

F(T,..,,){e9 ' ^+ '( )/(.T, y)} = e^^^^+'^y^F[T,,,,,+xhg'{x)+xy'h'{y)]f{x, y)

(2,5)

It may be noted that (2.4) can also be written as

i^(T,,,,,)Ki/'V(2:,y)} = 2:VF(T,+,+^,,,,,)/(:r,y) (2.6)

48

The analogues of Leibnitz formula for the opeartor Tk r v are

Ti;^,Jx'y(x.y)vix,y)} = x' E L W ; > ( ^ , 2 / ) } W , yu{x.y)} (2.7) n r=0

ri

r=0

??, T;:^^^^{y^Mx.,yHx.,y)} = y^" ^ [ I W , > ( . x , ?/)}{T;^,,,^(X, ^)} (2.

n

(2.9) r=0 V ^

?-=0 \ ' /

(2.10)

Here (2.10) can also be written as

T;:^,^,^{xVu(x,y)v{x.y)}=x'y' £ h{Tl;:^v{x.y)}{T^,, ,^v{x.y)} r=o yj

(2.11)

7T:.,,{xyt/(T,y)i;(x,y)} = xv t h{n;::A^,y)}K,r.M^,y)} (2.12)

Tl,^^^{x"y^^u{x,y)vix^y)} = x V E h {r^-:,.,^(x,i/)}{r^^,,,,^n(i;,y)}

(2 13)

Here (2.12) can also be written as

49

" n. n{x'^yl'u{x,y)v{x,y)} = x"y" E ( j W . ^ X ^ , ?/)}{^«+/V,,/^(^'^)}

r=0 (2.14)

From (2.7-14), we also have

.zT, — rr-^o^-i k r zT, e^-^''''-v{x''u{x,y)v{x,y)} = x'^e"^-^''''•yv{x,y)e^^''-'-v u{x,y) (2.15)

.^7V,?/{i^^j(x,T/)u(x,y)} = y^e^^^'-^-?/^j(x,y)e^^^^^'''/ii(x,i/) (2.16)

e^^(\^--^j{x^y^u

e^^^^'^^y{x^y^"u

e^^i^'^T'V{x''y^'u

.zTi ^^,y[x"yf\i

e^'^''^'''V{x"y^u

J„,k-zTkr zn x,y)v{x,y)} = x^^y^'e^-^''-^'-vv{x,y)e^'^^"-''-y u{x,y) (2.17)

.k^.h^zT' zTk x,y)v{x,y)] = xhf e^-^'^^'^^^yv(x,y)e^^^^-"^y u{x,y) (2.18)

,k,,k'zTk x,y)v(x.y)] = x V ' e ^ ''''' ^ (a ,2/)e ^^"•''••/ ^^(2:,?/) (2.19) T2A

S',.k^zT, x,y)v(x,y)} = .T^>7^^e - ' '- vz;(a:,?/)e' ^o,r7; ^^(x,^) (2.20) .A^ZTQ

li.zTi zTfi x,y)v{x,y)} = x''y^^e^^^+''^^^yv{x,y)e^^l^^''^yu{x,y) (2.21)

X, — rr(^ n fi O^-^k .1 . zTr, ,y)v{x,y)} = x''y'^e^-^'^"-'-yv{x,y)e^-^"+ft-''^yu(x,y) (2.22)

From (2.2), we have

ao^Toi.

e^

and hence,

exp(- t T21,

^ y ^ {l-xytY+'

the general formula

,,,„){.T^",/V(x^/)} = (^.

j+/^

n+,i f\

vi

2

— xyi' 1

(2.23)

i ] — xyf j

(2.24)

50

It is easily verified that

T.^^rr2kr.y{^'''-''y''-"f{x\,y')}

x"'y"'{l + ty'+"-'f{x\l + /.), r (1 + t)} (2.25)

rpn p ai,a2,- • • ,cir',xy

= 2"(^)„x^/,+,F,+i tti, a2, • • •. a,-, /c + n: xy

(2.26)

exp(-t T2k.T.ii) rF, ai,a2.---, ttr'.xy

-k l~xyt)-\F, ai, a 2 , • • • ) OT'I

hi.h2,--- .,b,]

xy 1 — xyt (2.27)

-^ 2k.r.y ^

a

id) e);( / )

2 2 = 2"(A:)„3:"z/"F

Q),k + n

(6);(c) 2 2

(2.28)

51

where

F

{a)

x,y = EE (2.29)

and a symbol of the type (a) stands for a sequence of parameters ai.02,- • • ^aA

etc. Thus l{b)]r means {b])r, (62)/, • • • (^;j)r etc.

exp{-tT2kj,y)F

e); if)

2 2

1 - x-gty'T

fi) (b). {c)

(d)

e)\{f)

2 2

1 — ryf ' 1 — ryl

^F,

Also, from (2.2), we have

a i , a 2 , • • • , flr! 2^ T2/f 17/

bi,b2,---,b,; {x'V'}

(2.30)

„2«,,2/i = :r^^^y^^+iF, a i , Q2, • • • ) 0,r; A" + O + /^; T?/t

^ i , ^ 2 , - - - , ^ s : (2.31)

-Fs a i , a 2 , • • • ,a r ; 2^ F2/i-,T,v

& i , ^ 2 - - - - , & s ;

2fS,2/:(^-r2-7/2 _ ^2f> 2/:( '^^ ( ^ i' ( V

r! s\

52

X r+lF^ aj. a2, • • •, CLr; k -\- a + P + r + s: x'^yH

(2.32:

If 1

-l^ k r ij

1

is the inverse of the operator Tr^k.y, then

(-1)" rpri -^ 2k,T,y

,,,, ^ 2 " ! + Q' + /3 , T ^ 2 " - ^ - 2 / i -

T, |(/c — 2) iogxlogi/+ iogxyj =

k.T,y X y

(2.33)

(2.34)

/ d d\ / d d' Again, let Sjjj^z = yzll + y~ + z-—\ , T .,,. = t(jx (/c + i(j— + x~

tlien

^ k,w,r I \ W X

( - ! ) " ( / + 7 + (5)„ ^27+«22^+"

I + a + (3 — k)r, w 2n+nj.2li+n (y + (3^k-\,k-2,

and hence, we have

•i^s

a i , 0 2 , • • • , a , u,;^, , u,r ,

bi,b2r--,b, ;

5*2/, (7,2

T,

2.35:

^2«3.2/y r+1 i^s+1

a i , Q 2 , • • • jCir,^ + 7 + (

61,62,- • • '^^5 1 + CK + /5 - ' '

- ^ ( / Z

ifjrc 2.36)

In particular, we have

53

l~t S' -n~ i~l+k

21,y z

Ti k.u) T

l~t: S~ 2I,v,z

T, k w) T

y^^z'' yj20y.2li W^"X^^^

y2n^26

y2j^26

y2,^26 1-f

tyz W X^ \ iVT

c,l + -f + 8

'l-"l~-6

(2.37)

iFi -tyz

1 + CX + (5 -k wx

2.38:

Also we have

exp 2a„.-2a-

2T, 2k,T,y J {x-"'y-'"}

no+fi—k li—o—k ^,(y~ft—k y"~l'^'ril + Q+P-k)J, a+li-k xy^

(2.39)

where a + /3 — /r is not a negative integer.

54

CHAPTER VI

SOLUTION OF MISCIBLE FLUID FLOW

THROUGH POROUS MEDIA IN TERMS OF

HYPERGEOMETRIC FUNCTION QFI

ABSTRACT: In this chapter a solution involving hypergeometric func­

tion o-Fi has been obtained for miscible fluid flow through porous media.

Here, the phenomenon of longitudinal dispersion has been discussed, con­

sidering the cross-sectional flow velocity as time dependent in a specific

1. INTRODUCTION: In the present chapter,\{eliavg4iscua^ed^a spe­

cific problem of miscible fluid flow through homogeneoiis porous media

with certain assumptions have been discussed. The mathematical formu­

lation of the miscible fluid flow through porous media gives a non-linear

partial differential equation, which is transformed into ordinary differen­

tial equation and reduced to a particular case of generalized hypergeomet­

ric equation (see Rainville [141]) by changing the independent variables

using subsequent transformation. Finally, one get a general solution in

terms of hypergeometric functions QFI.

The hypergeometric dispersion in the macroscopic outcome of the

actual moment of individual tracer particle through the pores and various

physical and chemical phenomenon that takes place within the pores.

This phenomenon simultaneously occurs due to molecular diffusion and

convection. This phenomenon plays an important role in the sea water

intrusion into reservoirs at river mouth and in the underground recharge

of waste water discussed by Scheidagger [148]. Here the phenomenon

of longitudinal dispersion which is processed by which miscible fluid in

laminar flow mix in the direction of flow.

Several authors have discussed this problem in different view point,

namely Scheidagger [148] and Bhathawala [7].

2. MATHEMATICAL FORMULATION OF PROBLEM: The

equation of diffusion for a hquid flow through homogeneous porous medium

without increasing or decreasing the dispersing material, is given by

div Dp grad

where D is the dispersion tensor, C is the concentration, u is the pore

velocity and p is density of the mixture.

The longitudinal dispersion phenomenon in a porous media is given

by

— + u~ ^ D,^^ (2.2) dt dx dx^

101

where v is the cross-sectional time dependent flow velocity.

DL is the longitudinal dispersion coefficient and boundary conditions

are

C{0,t) = Co, Cmt) = Ci (2.3)

where CQ is the initial concentration of the tracer, Ci is the concentration

aX X ~ L. Consider

I - X and ^ = T (2.4)

then equation (2.2) and (2.3) can be written as

and boundary condition become

C(0,T) - Co and C(l ,T) = Ci (2.6)

Assuming that concentration is expressible in the separation of vari­

ables form as

C(X,T) = F(rj).H{T). (2.7)

where

T] = - ^ , H{T) = T and ry G [0,1] (2,

102

then equation (2.5) becomes

D,F"{r]) + D2F'{r])-F{r}) = 0 (2.9)

where Di = D^. D2 = l-j - L with boundary conditions F(0) = CQ,

F{a) — Ci, wliere

« = ^ (2.10)

Equation (2.9) can be written as

F"{rj) + {a + bi])F'{r]) + dF(i]) - 0 (2.11)

where

a = - A , 5 = _ i ^ a n d d = - J ^ (2.12)

3. G E N E R A L SOLUTION: Substituting ^(77) = V{x), x ^ a + brj.

(2.11) and (2.10) reduces to

V" + lxV'{x) + ^V{x) = 0 (3.1) b b^

with boundary conditions

V (0) = Co and V{(y) - Ci (3.2)

Substituting V[x) — F(r) , 2r = -\'^^ then equation (3.1) reduces to

,y"(,) + Jl _ ,) K'(r) - | l /{r) = 0 (3.3)

103

with boundary conditions

Co, when rj = 0

C]. when rj = a and a Vt

Equation (3.3) is a confluent hypergeomctric equation and solution of

this equation is given by (Rainvihe [140]

V = iFi d 1

26' 2' '' 3.5)

Substituting r = 2t, V = e^wit), b = 2d in (3.3), we get

d , tw' w — tw — 0 (3.6)

and one solution of the equation is given by

w = e \FI d d

26' 6' ^

on changing the independent variables to y ~

reduces to nd^w (d l\ dw

and corresponding boundary conditions are

(3.7)

h^ in (3.6), then (3.6)

0 3.8)

d w 1664

a + hr]Y exp d

262 a + hr])'' Co at ?7 = 0 (3.9)

104

f £ 11664 exp

_d_ 262

a + brjY C'l at 77

one of the solution of equation (3.8) is given by

d 1 w; = 0^1

-' 2 6 ^ 2 ' ^

cf, where a = —^

(3.10)

(3.11;

the general solution of (3.8) is

w = A QFI d 1 " 26 + 2'^.

+ B{iyp^^^) oFi ' 3 d 1

"26'^. (3.12)

where A and B are arbitrary constant. Therefore solution of the equation

(2.11) is written as

F{r]) ~ exp

+B

l+T]f

2Dj A,Fi

l + ?7) 4D? )Fi

3 1 /1 + 7? (3.13)

using the boundary conditions as in (2.10), we get the value of constants

A and B as

A exp (^) {{A,)C,il + «)2( +2n) _ ^^(^^) ^^P(;L _

a

(^i)(53)( l + a)2(i+2.)_(^2)(Bi (3.14)

5 = exp ( 2 ^ ) {{B2)Ci exp(l + c>)2 - CoB^

^f'^"\AiB2{l + aY^'^'^^ ~ A2B, (3.15)

105

where

A, )i^i

M = (]Fi

3 1 /1 + a) - 2 + ^ ^ ^ ^ ^

3 1 (I)

Bi = oi^i

5, )i^i

1 1 /1 + a)^ -' 2 ~ ^ ' 4 ^

1 1 / 1 )

2 ' 4 I i^.

Hence with the help of equation (3.13), (2.8) the concentration (c) is

obtained.

4. CONCLUSION: We have discussed a specific problem of miscible

fluid flow through porous media under certain assumption and obtained

a general solution in terms of hypergeometric function o-f i- The solution

(3.13) represents the desired analytical expression for concentration and

result has physical significance which also satisfies the boundary condi­

tions.

106

CHAPTER V

ON A GENERAL CLASS OF POLYNOMIALS

OF THREE VARIABLES L(f' ' '* ' ' )(a?, y, 2)

ABSTRACT: This chapter deals with a study of a general class of poly­

nomial LI^^''^'^^ ^''\x,y, z) of three variables which include as particular

cases various polynomials of one, two and three variables studied earlier

by a number of mathematicians. Prominent among them are the polyno­

mials L"'^\T) due to Prabhakar and Rekha, L|f ^\x, y) due to S.F. Ragab,

Z"{x;k) due to Konhauser. L\^{T) due to Laguerre Z\"'''^^'\x.y.z]k.p,q)

due to M.A. Khan and A.R. Khan and three and m variables analogues

of Laguerre polynomials due to MA. Khan and A.K. Shukla.

Certain generating functions, a finite sum property, integral represen­

tation Schlafli's contour integral, fractional integrals, Laplace transform

and a series relation for this general class of polynomials have been ob­

tained in this chapter.

1. INTRODUCTION: In 1972, T.R. Prabhakar and Suman Rekha

[133] considered a general class of polynomials suggested by Laguerre

polynomials as defined below

" *• ' n! tnk\T(ak + l3+V ^ '

and proved certain results for these polynomials. Later, in 1978, they

[134] derived an integral representation, a finite sum property and a se­

ries relation for these polynomials. Evidently, if a = fc, a positive inte­

ger, then L"''\x^') = Z>^{x] k) which is Konhauser's polynomial and more

particularly Ll'^\x) = Lll^\x) where L\j^\x) is the generalized Laguerre

polynomials.

In 1991, S.F. Ragab [135] defined Laguerre polynomials of two vari­

ables U^-^^\x,y) as follows:

L^'^^'\x. y) = £(!i±^+ ^)r(^ + ^ + ^ nl

r^o r! r ( a + n - r + 1) r(/5 + r -f 1) ^ ' ^

where Ll^^\x) is the well known Laguerre polynomial of one variable.

It may be noted that the definition (1.2) for L^^^ ^'^'> (x, y) is equivalent

to

" ^ " ' ^ ^ " (n!)2 hhr\s\(a + l).{P + l \ ^'''^

where n is any positive integer or zero.

In 1997, M.A. Khan and A.K. Shukla [73] defined Laguerre polyno­

mials of three variables L "- '' )(,x, y, z) as follows:

80

L^y^^\x.y.z) = i^^:3li/l±lMl±lk

rh h h) r\ s\ k\{a -f 1),{P + 1),(7 + 1 ) , ^ •

They established various generating functions and integrals represen­

tations for these polynomials. In 1998, they further generalized these

polynomials to m-variables [74]

Recently, in 2002, M.A. Khan and A.R. Khan [100] defined and stud­

ied three variables analogue of Konhauser's polynomial as defined below:

toh h r\s\j\T{k]^-a + \)T{ps + (3 + l)T{qT + ^ + l) ^ '""'

which ioi k ~ p ^ q = \ reduces to (1.4)

Motivated by the above investigations the present paper is a study of

a general class of polynomials of three variables L^^''f'^^'^''^'''\x,y,z) sug­

gested by (1.1), (1.2) and (1.4). The polynomial L[,"'/^ ' ' ''' (.x,?y, z) yields

in particular cases the above mentioned polynomials (1.1),(1.2),(1.4) and

1.5).

81

2. THE POLYNOMIAL L^f' ' '- ' '/^^Cx,!/, ;s): The polynomial

l^i^(^J^'i^^'^-fj)(^x,y,z) is defined as follows:

r{oji,iAx,0(^ , ^\ - r ( ^ n + /3 + 1) r (7n + (5 + 1) r(An + /i + 1) 7Z! IP

i ( , sM) ,t o T\s\j\T{(y3 + /5 + l ) r (7s + ^ + l)r(Ar + M + 1) ^ ' ^

PARTICULAR CASES

i) Putting a = 7 = A = 1, (2.1) reduces to

.t^)sti, ,t^) r ! s ! j ! ( ^ + l ) , ( 6 + l ) , ( ^ + l ) , ^ •

which is a three variables analogue of Laguerre polynomials L[j^'^-''^ {x,y, z)

due to M.A. Khan and A.K. Shukla [73].

ii) Putting a = k, ^ ^ p and X = q^ where k./p and q are inte­

gers and replacing x by x}\ y by 2/ and z by z'' in (2.1), we obtain

Z\l^'^''^{x^y.z-.k.p.q)^ a three variable analogue of Kounhauser's polyno­

mials (1.5) due to M.A. Khan and A.R. Khan [100] which on further

putting y — z = {) gives Konhauser's polynomial Z'^{x\ k).

in) Putting a = 7 = A = 1 and z = 0 in (2.1), we obtain

\n\

^ f^Y:^- {~n)s+jTMf

which is i i : t _ / ^ L^''\x,y), where Lf'^)( .T, ?/) is a two variable analogue TV.

of Laguerre polynomials due to S.F. Ragab [135] which in turn reduces

to Laguerre polynomials for a: = 0 or ^ = 0.

3. GENERATING FUNCTIONS: The polynomial 4"'^^'^'^'^'/')(,T, y, z)

have the generating function indicated in

e'$(», ^ + 1 ; -xt) $(7, (5+1; -yt) $(A, ^LX+I; -zt)

~ „^o r(c.n + /5 + l ) r (7n + 6 + 1) r(An + M + 1) ^ • '

where ^{a,l3]z) is the Bessel-Wright function [Erdelyi [23], p.211].

For an arbitrary c, we easily obtain the following generating function

for (2.1)

^ {v\)'\c)„L[;''^^'^^^-'^'i'\x,y,z)t^

; a r(t tn + / + l ) r (7n + b+\) V[\n + /i + 1)

I \-( ( i ~" ^ "• ^ ~'^^ ^ Cl 0

83

where

.cj,+s+? x^ y' z' EU,ci,ejAx. y, z) ^ E ^ , ^ , , ^^^^ ^ ^^ ^^^^ ^ ^^ ^^^^ ^ ^^ (3.3)

It may be regarded as a three variable analogue of Wiight's function

E'^i^{z) defined by

0° (r) z'

K,>U) = T~^'~~, (3-4) =0 r (a r + h) r\

If we replace thy ~ and let I c |-^ oo in (3.2), we obtain the generating c

relation (3 I]

Before proceeding to obtain more general sets of generating functions

for (2.1), we introduce here Wright's type triple and quadruple hyperge-

ometric series respectively as follows:

^M) a.A) v.{h,B)-{h',B')-{h\B"):{d,D)-{d\D'):{d!',D") ;

.r, y. z (/, F) :: {g, G): {g'. G')- ( / , G") : (K H); (h', H'): {h\ H'

CO CX) OO T{a + A{m + n + p)}T{h + B{m + n)]T{h' + B'{m + p)}

h o h h r { / + n^-^ + n + p)}r{g + G{m + n)}r{g' + G'{m+p)]

r{b" + B'{n + p)}rid + Dm)rid' + D'n)T{d" + D"p) x'" ^" z^'

T{g" + G"{n + p)}r("/i + i 7 m ) r ( F + if'r7)r(/z^' + i:f>] m!?7!p! ^ ' -

84

* (4) a, A) ::: (b, B)-(b'.B'),{b",B"):{h"\D"') :. {d,D)-{d\D'Y{d".D"):

(/, F) ::: (,g, C); (^', G'); ( / , G")- {g"\ G'") :: (/7, H): {h\ H')- (h". H"):

{d"\D"'):{d'\,D'^);{d\D'') : {e. E):{e'. E');{e", E");{e'", E'") ;

(/1-, i f - ) ; (/7-. F - ) ; (/^^ i/^') : (z, / ) ; (/, / ' ) : {i'\ I"); {^"'. I w,x,y,z

Tlft\

oo oo oo oo _ T{a + A{m + 77 + fe + 7)}r{6 + ^ ( m + ?7 + fc)}

,„=Or.=ofr=oj=o r { / + -F(m + 7? + /c + j )} r fe + G(?7? + n + fc)}

V{b' + ^ (?77 + n + 7)}r{&^ + B"(m + k + ])}T{b"' + B'^n + k + j)}

V{g' + G'im + n + ])]T{g" + ^"(m + A: + l)}T{g"' + G""(r7 + A- + j )}

r{(i + D{m + ri)}r{c/^ + D'{m + fe)}r{(i^^ + L'^X^ + ])}^{d"' + i^'''(n + k)}

V{h + //(?77 + n)}T{h^+ H'{m + A-)}r{/?'' + H"{m + ?)}r{/i- + //-(?i + k)}

r{d'' + D' {n + j)}T{d' + D^ (fc + j )}r(e + Em)r(e ' + E'n)

r{/i'^ + i/ ^(n + j)}r{/i^ + H^{k + j/)}r(7 + /r??)r(z' + rn r(e" + £^"A:)r(e'" + E'"]) w"' x'' y^ z^

T{i" + I"k)Ti^F+r"]) m\n\k\ ]( (3.6)

Now for arbitrary p and g, one can easily obtain the following more

general generating function for (2 1):

^ {n\fV{j)n-^q) L^^'''^''^^ ^''\x,y,z) f

^tl) T{an + P+1) r(jn + 6 + 1 ) 1 ^ + fx+l] ^ ^ ( 4 )

(q.p) : : : ^ ; _ ; _ ; _ : : „ ;

- ) „5 - 1 - ) - •

-TI . —yt, —zt. t

(/3 + l , a ) ;((5 + l,7) ; (A + 1,A) ;. (3.7)

85

Interesting special cases of (3.7) are as follows

^0 r(77? + 5 + 1) T{Xn + /i + 1) "

( / 3 + l , Q ' ) : : : ^ ; _ ; „ ;

—xt, —yt, —zt, t j _ ; _ : ( / 3 + l ,») ;('5 + l,7) ;(/" + l-A) ;„ ;

(3.8;

°2. (n!)2 4"''^'^''^'^'/^)(x,?/,2)f

r(A?? + A/ + i ^(4)

(/?+l,c.)(^ + l ,7) :::-;_:

_ ) _ 5 _ ; _ • • _ 5 _ 5

-2: , —yt, —zf^ t

_;_;_:_ : ( / 5 + l , » ) :(<5+l,7) :(/i + l,A) ;

(3.9)

0 0

E(n!)2L(^^^^'^'^'^'")(:r,?/,z)f r/=()

^(4)

( /?+l ,c . ) (^+l ,7)( /7 + l,A) - ) —; — ) — )

—a;f, —yt, —z/,/-

: ( / 3 + l , « ) ; ( ^ + l , 7 ) ; ( M + 1 , A ) :

3.10)

Yet other generating relations of interest for (2.1) are as follows:

oo

E (nl)^ L^:'-^^^-^^-"^{x,y,z)f

^) r(pn + rj)r{(yn + /? + l)r(7n + 6+ l)T{Xn + fi + T

= ^ (4) _ ) — ) — J — ) — 1 .

iv.p) ••••:-

—xt, ~yf. —zt,t {(3+l,a] ;((5+l,7) ;(/i + l,A) ;

(3.1i;

(n!)2r{pn + q) Ll^'-'^'-^'-^^^-i'^x, y, z) f y — „t o ^{pn + ?7)r(m7 + /3 + l)r(7n + 5+ l)T{Xn + //

= ^(4)

(?],/>) ::: -1 — ) — • — ) — )

—xt. —yt. —zt.i ( /5+l ,a) ;((5+l,7) :(/ i+l,A) ;_ ;

In particular, when Q = ^==\ = p = p— l^ (3.12) reduces to

(3.12)

E

^pW

Tj . . . _ , _ ) _ . _ . • _; _ 5 _, ^. _ 5

~xt, —yt, —zt, t /5+l ;5 + l;// + l;_ ;

(3.13)

87

where L^'^ ^'^\x,y. z) is Laguerre polynomials of three variables due to

M A Khan and A.K. Shukla [73] and F^'^^liu. x.y.z] is a quadruple hy-

pergeometric function

Further, putting a = k.'y=p, X = q where k,p and q are positive

integers, replacing x by x\ y by y^' and z by z^ and putting /? = 1. /) = 1

and replacing q by A in (3.12). we get

g {n\)HX)„Zi;^'''\x,y,z;k,p,q)t- ^ ^^,^ A : : : : - • - ) - ) - • • - ) - ) - ) _) --)

V -: J J J - ) _ 7 _ • _ j _ ; .

-xt, —yt, —zt,t /\{k,p + l) ;A(;9,<5+1) ;A((7,^ + 1) ;_ ;

(3.14)

4. A FINITE SUM PROPERTY OF ^(^"'^'^'^'^''^H^^y^^): I is

easy to derive the following finite sum property of the polynomial

l{<^J^n^^^i'){yo:r,wy.wz]

{k\fr(m, + /5 + l ) r (7n + 6 + l)r{Xn + /x + 1

n\y{n - k)\T{ak + /3 + l)r{-fk + (5 + l)r{^k + fi+l)

(4.1

PROOF OF (4.1): The generating relation (3.1) together with the fact

that

= e^'-""^'e'"'^{a, /3+1: -x{wt))^^, 6+1: -y{wt))^{\, / i+1; -^z{wt))

yields

- {l-wYt-\ / - (n!)2L(^Vi,7AA,/v)(3.^^^^) ^ . ^r

v«=o ^! I \v^() ^((yn + /? + l ) r (7n + 6+ l)r(An + A + 1]

from which, on comparing the coefficients of T on both sides, we get (4.1).

For a = '^ = \ = 1, (3 replaced hy a, 6 replaced by (3 and /;, replaced

by 7, we get the correpsonding result for Laguerre polynomials of three

variables L^^'^^'"^\x,y,z) defined and studied by M.A. Khan and A.K.

Shukla [73]. Some other particular case of interest are as follows:

i) For A = 1, /x = 0, 2 = 0, (4.1) reduces to

^ ^ fc! r((>n + / 3 + l ) r ( 7 n + ( 5 + l ) ( l - ^ < - V (,,,;,,,)

to in])in-ky.rio'k + P + l)rijk + 6 + l) ^' ^""'^^ ^ ^

which is a finite sum property for the polynomial Ll^^-^^'^-^\x,y) defined

and studied by M.A. Khan and K. Ahmad [110].

ii) For A = 7 = 1, /i = ^ = 0, T/ = 2 = 0, (4.1) reduces to

^" ^ ^"".t^o in^k)\T{ak + l3 + l] ^' ^""^ ^ ^ ^

89

which is a finite sum property for Prabhakar and Rekha's polynomial

L'^'f\x) which for » = 1 and (3 replaced by a gives

k=o {n~k)\[Q + l)k

where L]^\'wx) is a well known Laguerre polynomial.

iii) Putting a = 7 = /c, a positive integer, A = l , /i = 0, 2: = 0 and

(5 replaced by a, ^ replaced by /5, x replaced by x'' and y replaced by

y^\ we get the following finite sum property for two variable analogue

of Konhauser's biorthogonal polynomial Z^'^\x^y]k,p) studied by M.A.

Khan and K. Ahmad [79];

Z'j^'^\wx,wy,k, k)

~- £^^)-<;;^;f;^:"^>"'>^^(x..,.) (4.4, r

iv) Putting Q' = 7 = A = /c;, a positive integer, /?, 8 and [j, replaced by a, j3

and 7 respectively and x.y and z replaced by .T'',T/''" and z^' nispectively,

one gets the following finite sum property for three variable analogue

Z'^''^'"^{x,y,z\k,p,q) of Konhauser's biorthogonal polynomial studied by

M.A. Khan and A.R. Khan [100]:

X w^''{l - w'y-^' Z; - - (.x, y , z- k, k, k) (4.5)

90

5. INTEGRAL REPRESENTATIONS: Using the definition of Beta

function it is easy to derive tlie following integral representation of

L„(a,/5;7, (5; A,/i)(a:.7/, z) (see Rainville [140]):

it IS n

( j^j\\T-xY'-'v\s~yf'h^\t-zy''-'L\^'^^'^'^^'\T:\y\z')dxdydz

T{an + /? + l ) r (7n + (5 + l)r(An + /i + 1)/^+/ '/+' ^^/^ V

Y{an + /? + /? 'T l ) r (7n + ^ T ^ + l)V(\n + /; + A/' +^0

using the integral (see Erdelyi et.al [21], pp.14),

2ismTizViz) = - l^^^\-tY"h-' dt (5.2) /oo

and the fact that

(l-x-y-zT = E S E ^-^f~r^ (5.3)

it is easy to derive the following integral representation for

L,,(a.J;-f,5;X.iJ.){x,y,zy.

iO+ /()+ /0+

/oo /cx3 /oo

—zi—wyY^diidvdw

-1

00 /oo /oo

r(Qn + /5 + l)r(7f7 + (5 + l)r(An + A + 1)

X LY'^^^^^''\x,y,z) (5.4)

91

we also have

f??')'^ /OO /OO /OO u'^v^w"e-"-'-'"

X Ll^'f^^^^^^''\xv'\yv^,zw^)dudvdw = {1 - x - y - zf (5 5)

6. SCHLAFLFS C O N T O U R I N T E G R A L : It is easy to show that

, , . . , w ^ r(OT + /3 + l)r(7/7 + 6 + l ) r (An + // + 1)

(0+) .(0+) .(0+) iu"v^w'-xw'-yv^~zin\^,,^.^.^, ^^^^^^ .g . -OO /-OO /-OO -^Q??+/Hl-j;T77+(5+l^A??+/J + l ^ ^

P R O O F OF (6.1): The R H S of (6 1) is equal to

rjan + /3 + l)r(772 + 6 + l ) r (An + /7 + 1) " ^"^-^ (-/7),+s+;:r^ ^^ z'

'n^y r=Os=o ?=o r ' 5' :;'

X - i - /^'""^r^^^^+^^+^^eVi/-^ /^ '^^- (^^+^+^)eVi ; -^ /^"'•\.-(^'^+/'+^)e"rfi^ 27T? ^ 00 27rz '"00 27rz ^-00

/? "n—rn—T — s V(an + /? + l ) r ( 7 n + (5 + l ) r (An + /; + ! ) " ' ^ ' "'^-^ (-r?

X x- y^ z^

r(Q'77 + /? + l ) r ( 7 n + ^ + l ) r (An + /i + 1)

using the Hankel's formula (see Erdelyi, A et al [21] 1 6(2))

1 "'"'e'r^rfi (62; r(2;) 27rz '"OO

92

finally (6.1) follows from (2.1).

For a = k, ^ = p X = q. where k,p and q are positive integers, /5, S

and /i replaced by a, [3 and 7 respectively and x, y and ^ replaced by

x^\ y^' and 2; respectively, (6.1) reduces to the corresponding Schlafii's

contour integral for three variable analogue of Konhauser's polynomial

Z"'^^'^{x.,y,z;k,p,q) obtained by M.A. Khan and A.R. Khan [100].

7. FRACTIONAL INTEGRALS: Let L denote the linear space of

(equivalent classes of) complex-valued functions f(x) which are Lebesgue-

integrable on [0,a], a < 00. For /(j') G L and complex number /x with

Re{n) > 0, the Riemann-Liouville fractional integral of order fi is defined

as (see Prabhakar [130], p.72)

I"f{x) = -—r l^^{x - ty"'\fit) dt for almost all x e [0, a] (7.1)

using the operator / ' ' , Prabhakar [131] obtained the following result for

Re(//) > 0 and Re(a) > — 1:

V'[x"Z':{x;k)] = r(fcn + a + l ) ^ ^ ^ „ , „ ^ ^ ^^2) i [kn + « + /,/ + !)

where Z"{x; k) is Konhauser's biorthogonal polynomial.

In an attempt to obtain a result analogous to (7.2) for the polynomial

( o,/:(;7,<5;A,/v) ^ ?/, z), wc first seck a three variable analogue of (7.1)

93

A three variable analogue of V may be defined as

r(0r(^)r(7y)

X 1^^ IJ j^{x - uf~\y - vY~\z - wf-\f{v,v, w)dudvdw (7.3)

putting f{x,y,z) = xPy^z''L[^^f^^^^^^^^i'\x'\y\z^) in (7.3), wc obtain

/• '''[.x' t/ z' Lj; ' ' ' ^'''\x'\y\ z^)] = V /-2

X — U 1^-1

r(0r(/^)r(/7) ^ o ^ X (2/ - i;)/ -~ ( - t ) - zi ^u i; ' Li';'f''^^^'^'"\i/\ v\ t//) rfw ^7; dw

r^^+fiyP+6^r,+,j ^j ^^ ^j

r(0r(p)r(/7) /o /o /o X Li;'^^'^^^"\x'T,y'^p\z^q^) dt dp dq

(by putting u = xt, v ^ yp and 7x' = ^g)

_ rjmi + P + l)T{in + S + l)T(Xn + /i + l)x^+^U/^^z'l+^'

^ " "^'-"^-^ ( -?2) ,+ s+j3: ^^ ?/ ^ z^'

X 1^ i"^+^\l - ) - (ft j^^p'"^\l - p) ^"Vp /^ g ''+ '(l - qf'Hq

_ V{mi + / + l)r(7n + ^ + l)r(An + /i + 1) :re+/ y/- + ^ +/

^ , , _ , _ _ , (-n),+,+,(x-0^ (7/ )- {z^Y

""r^ost^, ,=0 r!5!j!r(^.7+/? + C + l)r(75 + (5 + / + l)r(Ar + /7 + 7? + l

94

We thus arrive at

/e,/>,^[^/i^«^/v^(a./i;7,5;A,/.)(^a^ ^^, z^)] - r(Q-n + /3 + l ) r (7n + ^ + l )

r{an + /? + ^ + i)r(7n + (5 + p +1;

r(An + ^ + 77 + i) " [^ ^y ^ ^ ) [(•^}

8. LAPLACE TRANSFORM: In the usual notation the Laplace

transform is given by

'•C»

L{f{t) : s] = 1^" €-V{t) dt, Re(s - a) >0 8.1'

where / G L(0, R) for every i^ > 0 and f{t) = 0{e"'), t -^ oo. Using (S.L

Srivastava [16] proved

L{fZ'^{xt\k) -.s} g + i)A^„r(/3 + 1

s/ +i n! i^fr k+l-i'k

-n,Aik,P + l

_ A{k,Q-h 1 3

A'

8.2)

provided that Re{s) > 0 and Re(/3) > —1.

In an attempt to obtain a result analogous to (8.2) for the polynomial

j^{(iji;j,6;X,tj)^^^ ?/, z) we take a three variable analogoue of (8.1) as fohows;

L{f{u,v,w) :5],S2,S3}= 1^^ 1^^ 1^^ e CO /•CO /'CO

•,SiW~,S-2?'—.S3W; f{u,v, w)du d.v dw

!.3)

Now, we have

95

L{u^vf'w'^Ll:,'^'^^'^'^^^^'\xu'\yv^.zw^) : 31,82,53}

r{an + /? + l)r(7n + (5 + l)r(An + ^ + 1)

^ h^ s=o j^o H 5! j ! r (a j + ;5 + l)r(75 + ^ + l)r(Ar + / / + ! )

h /() /o r(an + P+ l)r(7n + ^ + l)r(An + /; + 1) " "^^"^-'' (-n),+,+^

(n!)M^^5r^sr rTos^) jt^) r\s\j\

^ r(aj + e + i)r(75 + /9 + i)r(Ar + 7/ + i) / x v / ? / V 2 ^^

We thus arrive at

L{i/i/'w'^Li''^^^^'^^^'^^^'^ixu",yv\zw^) : 81,82,83}

_ r(Q'n + /3 + l)r(772 + 8 + l)r(An + /i + 1)

-n , l , l , l ) : :^; j_:(e + l , a ) ; ( p + l , 7 ) ; ( ^ + l , 7 ) ;

x F X y z

: : J ^ ; _ : ( / 3 + 1 , Q ) ; ( ( 5 + 1 , 7 ) ; ( / 7 + 1,7)

• i S2 S j

;8.4)

In the special case ( = /?, p = , 77 = /i, (8.4) simplies to the elegant

form

L{u'^v^ni'L^;'^^^^^^^^''\xv'\yv\zw^) : 81,82,83}

96

r ^ n 3 »r)+ii+i -fn+6+1 A77+//+1 [ni) Si §2 S3

(8.5:

9. A SERIES RELATION F O R i:( «,/3;7, ;A,M)( ^ y^ ^y_ VVg ^^o^ ^ ^ j ^ ^

use of fractional differentiation operator D'^ defined by (see [158], p.285)

'^ ^ r ( A - / v + i)

where /i is an arbitrary complex number to show that

(9.1;

,;tor(/y | -n ) r (an + /? + l ) r (7n + ( + l)r(Ar/, + / H 1) ' '^^ ' ' "'^

= e'¥'^ fel • • _ ) _ ) _ •

( i^ , l ) : :_; j_:( /3 + l,a):(<5 + l,7);(A + l,A) ; -xt, —?/t, —zi

(9.2)

P R O O F : We can rewrite (3.1) as

00

e-'E {n])'^Ll^"^^'^'^-''\x,y.,z)f

„% r{an + /? + l ) r (7n + S + l)r(An + /i + 1

= $(» , /? + 1; - x t ) $(7, (5+1; -1/t) $(A, A + 1: -z^

97

which can further be written as

hi)ho k\ T{an + /? + l)r{jn + 6+ l)r(An + fi+r

c» CO oo

hk ojh nlkljl V{an + /3 + l)r(7A: + 6 + l)r(Aj + /x + T

Now multiplying both sides by f^ and apply Df ', we obtain

L H S = ^t''-' t (-l)'(ri!)'(^)«+^4"'^''^'''''^^H^,?/,^)^"^' r(zy) „Y^o k\ T(an +(5 + l ) r (7n + (5 + l)r(A/7 + /i + l)(z/;

r(i/) „ro {y)v T H + /? + i)r(7r? + ^ + i)r(An + / +1

X i-P i h + n; iv + n; -t]

r(i/) ^ „tl)(i^)„r(an + / 5 + l ) r ( 7 n + (5+l)r(An + / i + l

X iFi[r] -n;iy + n;t]

using Kummer's transformation).

98

Similarly,

R.H.S. = f-' Z

X

r(?] + n + k + j)

n\ k\ jl

{-xtn-ytYi-zty r{iy + n + k + j)T{Qn + P + l)T{yk + 8 + l)r(A.7 +/1+I

= ^i^-l^Jf(3) (^,1)

—xt, —yt, —zt (zv, 1) ::_;_;_: (/3 + 1, c.); (5 + 1,7); ( M + 1 , A) ;

which immediately lead to the result of (9.2).

For A = 1, /x = 0 and z = 0, (9.2) reduces to the result due to M.A.

Khan and K. Ahmad [110] for L\^'-f^^-^\x, y) and for A = 7 = 1, /i, = 5 = 0

and ?/ = z = 0, (9.2) reduces to the result due to Prabhakar and Rekha

[134] for U^;'\x).

CHAPTER VI

SOLUTION OF MISCIBLE FLUID FLOW

THROUGH POROUS MEDIA IN TERMS OF

HYPERGEOMETRIC FUNCTION o^i

ABSTRACT: In this chapter a solution involving hypergeometric func­

tion (]Fi has been obtained for miscible fluid flow through porous media.

Here, the phenomenon of longitudinal dispersion has been discussed, con­

sidering the cross-sectional flow velocity as time dependent in a specific

form.

1. INTRODUCTION: In the present chapter, we have discussed a spe­

cific problem of miscible fluid flow through homogeneous porous media

with certain assumptions have been discussed. The mathematical formu­

lation of the miscible fluid flow through porous media gives a non-hnear

partial differential equation, which is transformed into ordinary differen­

tial equation and reduced to a particular case of generalized hypergeomet­

ric equation (see Rainville [141]) by changing the independent variables

using subsequent transformation. Finally, one get a general solution in

terms of hypergeometric functions QFI.

The hypergeometric dispersion in the macroscopic outcome of the

actual moment of individual tracer particle through the pores and various

physical and chemical phenomenon that takes place within the pores.

This phenomenon simultaneously occurs due to molecular diffusion and

convection. This phenomenon plays an important role in the sea water

intrusion into reservoirs at river mouth and in the underground recharge

of waste water discussed by Scheidagger [148]. Here the phenomenon

of longitudinal dispersion which is processed by which miscible fluid in

laminar flow mix in the direction of flow.

Several authors have discussed this problem in different view point,

namely Scheidagger [148] and Bhathawala [7].

2. MATHEMATICAL FORMULATION OF PROBLEM: The

equation of diffusion for a liquid flow through homogeneous porous medium

without increasing or decreasing the dispersing material, is given by

div Dp grad — -div{«c) = — (2.i:

where D is the dispersion tensor, C is the concentration, u is the pore

velocity and p is density of the mixture.

The longitudinal dispersion phenomenon in a porous media is given

by

— + n — = Z ^ L ^ (2.2) dt dx dx^

101

where u is the cross-sectional time dependent flow velocity.

Di is the longitudinal dispersion coefficient and boundary conditions

arc

C(0,t) = Co, C{L,t) = Ci (2.3)

where CQ is the initial concentration of the tracer, Ci is the concentration

at ,T = L. Consider

I = X and -^ = T (2.4)

then equation (2.2) and (2.3) can be written as

dC BC ^ d^C + uL^^ = D L T ^ ^ (2.5)

and boundary condition become

C(0 , r ) = Co and C{l,T) = Ci (2.6)

Assuming that concentration is expressible in the separation of vari­

ables form as

C{X,T) = F{r^).H{T). (2.7)

where

V = - ^ , H{T) = T and rj G [0,1] (2.8)

102

then equation (2.5) becomes

DiF"{r]) + D2F'{Ti)~F{Tj) = 0 (2.9)

where Di = Di, D2 = i-j — L with boundary conditions F(0) = Co,

F{a) = Ci, where

^ = ^ (2.10)

Equation (2.9) can be written as

F"{r]) + {a + bri)F'{7]) + dF{r]) = 0 (2.11)

w here

3. G E N E R A L SOLUTION: Substituting F(7]) = V{x), x = a + br],

(2.11) and (2.10) reduces to

V" + lxV'ix) + ^V{x) = 0 (3.1) 0 0

with boundary conditions

1/(0) = Co and V(a) = Ci (3.2)

Substituting V{x) = V(r), 2r = - | x ^ then equation (3.1) reduces to

rV"{r) + [l-r) V'{r) - | l / ( r ) = 0 (3.3)

103

with boundary conditions

l ^ j - —(a + 677)2| = Co, when 77 = 0

1/j — — (a + 6a)^ > = Ci, when TJ = a and a = 1

Equation (3.3) is a confluent hypergeometric equation and solution of

this equation is given by (Rainvihe [140])

V = iFi d 1

26' 2' " (3.5)

Substituting r = 2t,V = e^w{t), b = 2d in (3.3), we get

tw + J w — tw = 0 6

(3.6)

and one solution of the equation is given by

w e-\Fi d d •

26' 6' ^ (3.7)

on changing the independent variables to y •

reduces to nd'^w (d l\ dw

^ d^ + U + 2 j d - '

and corresponding boundary conditions are

0!

-t"^ in (3.6), then (3.6)

0 (3.8)

( £ w I -r-r7T(a + brj)'^ [1664 exp 262

a + brjy Co at ?7 = 0 (3.9)

104

w I— [a + brj)'^ > = exp 262

[a + 6?])' Ci at ?7 = », where a 1

(3.10)

one of the solution of equation (3.8) is given by

w = (jFi d 1 •

26 + 2 ' ^ (3.1i;

the general solution of (3.8) is

w = AQFI d 1

" ' 2 6 ^ 2 ; y + Biiyp-^^) oFi 3

2 26' ?/ (3.12)

where A and B are arbitrary constant. Therefore solution of the equation

(2.11) is written as

F(r]) = exp

+B

l + T])'

2Di

( 1 4 ^ +7?

A,Fi

)Fi

1 1 /1 + ry

3 W l + J?' (3^3)

using the boundary conditions as in (2.10), we get 1,he value of constants

A and B as

exp L}-) {{A,)C,{1 + a)2(i+2.) _ ^^^^^^ ^^p(^ _ ^)2-A

{A,){B,){l + cyf^'+^-)-{A2)(B, (3.14)

5 = e ^ p ( 2 ^ ^2)Ciexp(l + c ) 2 - a ) 5 i / ^ x2(l+^)

\2Di_ /lii32(l + a)2(i+2")-A25]

(3.15

105

where

A,

A,

)Fi

)Fi

3 1 /1 + al

3

' 2 1 (r

n: 4 \D,

Bi = oFi 1 1 fl + a)

4 D L

Bo )Fi 1

-' 2 1 / 1

n: ^\DL,

Hence with the help of equation (3 13), (2.8) the eoneentraiion (c) is

obtained.

4, CONCLUSION: We have discussed a specific problem of miscible

fluid flow through porous media under certain assumption and obtained

a general solution in terms of hypergeometric function QF] . The solution

(3.13) represents the desired analytical expression for concentration and

result has physical significance which also satisfies the boundary condi­

tions.

106

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39. KHAN, M.A. Certain fractional q-integrals, q-derivative,

Nanta Mathematics 7 (1974), 52-60.

40. KHAN, M.A. On q-Laguerrepolynomials, Ganita, 34 (1983),

111-123.

41. KHAN, M.A. On a q-Laguerre polynomials and their

characterizations, Ganita, 34 (1983), 44-52.

112

42. KHAN, M.A.

43.KHAN, M.A. AND

KHAN, A.H.

44. KHAN, M.A.

On a characterization q-Laguerre polynomials,

Ganita, 34(1983), 111-123.

Fractional q-integrals and integral

representations ofbibasic double

hyper geometric series of higher

order, Acta Mathmatica Vietnamica, 11 (2)

(1986), 2345-240.

q-analogues of certain operational formulae,

Houstan Journal of Mathematics, U.S.A. 13

45. KHAN, M.A. AND

KHAN, A.H.

46. KHAN, M.A. AND

KHAN, A.H.

(1987), 75-82.

An Algebraic study of a class of integral

functions. Pure and Applied Mathematika

Science, 29 (1-2) (1989), 113-124.

A note on mixed hyper geometric series^ Acta

Mathematica Vietnamica, 14 (1) (1989), 95-

98.

47. KHAN, M.A. AND

KHAN, A.H.

On some characterizations ofq-Bessel

polynomials. Acta Mathematica Vietnamica,

15(1) (1990), 55-59.

13

48. KHAN, M.A. AND

KPiAN, A.H.

49. KHAN, M.A. AND

KHAN, A.H.

50. KHAN, M.A.

51. KHAN, M.A.

A study of certain bibasic fractional integral

operators, Pure and Applied Mathematika

Science, 31 (1-2) (1990), 143-153).

A note on q-Lagranges polynomials, Acta

Ciencia Indica, 17 M (3) (1991), 475-478.

On a calculus for the T^^-operator,

Mathematica Baikanica, New Series, 6 (Fasc.

1(1992), 83-90.

On Generalized Fractional q-integrals,

Mathematica Baikanica, New Series, 6 Fasc.

52. KHAN, M.A.

53. KHAN, M.A.

54. KHAN, M.A. AND

2(1992), 155-163.

Transformations of discrete hypergeometric

functions, Mathematica Baikanica, New

Series, 6 Fasc. 3 (1992), 199-206.

Transformations of generalized fractinal q-

integrals, I. Mathematica Baikanica, New

Series, 6 Fasc. 4 (1992), 305-312.

Unification of q-Bernoulli and Euler

KPIAN, A.H polynomials, Pure Mathematica Manuscripts, 10(1992-93), 195-204.

114

55. KHAN, M.A. On some operational generating formulae,

Acta Ciencia Indica, 19M (1) (35-38.

56. KHAN, M.A. On certain transformations of generalized

Fractinal q-integrals JI, Mathematica

Balkanica, New Series, 7 Fasc. 1(1993), 27-

33.

57. KHAN, M.A. (p-q)-analytic functions and discrete bibasic

hypergeometric series, Math. Notae,

Argentina, 37 (1993-94), 43-56.

58. KHAN, M.A. Discrete hypergeometric functions and their

properties. Communications, 43 Series Ai

(1994), 31-412.

59. KHAN, M.A. Bibasic analytic functions and discrete

"bibasic" hypergeometric series. Periodica

Mathematica Hungarica, 28 (2) (1994), 39-

49.

60. KHAN, M.A. On some operational representations ofq-

polynomials, Czechoslovak Math. J. 45

(1995), 457-464.

15

61. KHAN, M.A. Expansions of discrete hyper geometric

functions, I. Journal of the Bihar Math. Soc,

62. KHAN, M.A. AND

NAJMI, M.

16(1995), 29-46.

On complex integrals of discrete functions,

Math. Notae, Argentina, 38 (1995-96), 113-

135.

63. KHAN, M.A. AND

KHAN A.H.

64. KHAN, M.A. AND

AHMAD, K.

65. KHAN, M.A.

66. KHAN, M.A.

Transformations of mixed hypergeometric

series, Demonstratio Mathematica, 29 (3)

(1996), 661-666.

On some rodriguews type fractional derivative

formulae, Demonstrio Mathematica, 29 (4)

(1996), 749-754.

Expansions of discrete hypergeometric

functions II, Acta Ciencia Indica, 22M (2)

(1996), 146-156.

Complete FORTRAN program of certain

important q-integrals, Acta Ciencia Indica, 22

M (2) (1996), 157-158.

67. KHAN, M.A. On some mathematical observation, Acta

Ciencia Indica, 22M (2) (1996), 167-170.

116

68. KHAN, M.A. AND

NAJMI, M.

69. KHAN, M.A.

70. KHAN, M.A. AND

SHUKLA, A.K.

71. KHAN, M.A.

On generalized q-Rice polynomials, Acta

Ciencia Indica, 22M (3) (1996), 255-258.

A study of q-Laguerre polynomials through the

Tkqx-operator, Czechoslovak Math. J. 47

(122) (1997), 619-626.

Transformations for q-Lauricella functions, J.

Indian Math. Soc, 63(1) (1997), 155-161.

On a algebraic study of a class of discrete

analytic functions, Demonstration

72. KHAN, M.A.

73. KHAN, M.A. AND

SHUKLA, A.K.

Mathematica, Poland, 30(1) (1997), 183-188.

Discrete hyper geometric functions and their

properties. Demonstration Mathematica,

Poland, 30(3) (1997), 537-544.

On Laguerre polynomials of several variables.

Bull. Calcutta Math. Soc. 89 (1997), 155-164.

74. KHAN, M.A. AND

SHUKLA, A.K.

A note on Laguerre polynomials of m-vartables.

Bulletin of the Greek Math. Soc. 40(1998),

113-117.

117

75. KHAN, M.A. AND On Hermite polynomials of two variables

ABUKHAMMASH, G.S. suggested by S.F. Ragab 's Laguerre

polynomials of two variables, Bull. Calcutta

Math. Soc. 90(1998), 61-76.

76. KHAN, M.A. AND On a generalization of Bessel polynomials

AHMAD, K. aP, suggested by the polynomials Ln (x) of

Prabbhar and Rekha, J. Analysis, 6 (1998),

151-161,

77. KHAN, M.A. AND On Langranges polynomials of three variables,

SHUKLA, A.K. Proyecciones, 17 (2) (1998), 227-235.

78. KHAN, M.A. A discrete analogue of Maclaurin series, Pro

Mathematica, 12 (23-24) (1998), 67-74.

79. KHAN, M.A. AND On a two variables analogue of Konhauser 's

AHMAD K.biorthogonalpolynomial Z„" (x;k), Far East

J. Math. (FJMS), 1(2) (1999), 225-240.

80. KHAN, M.A. On a generalization of Z-transform, Acta AND

SHARMA,B.S Ciencia Indica, 25M(2) (1999), 221-224.

81. KHAN, M.A. AND SHUKLA, A.K.

On some generalization Sister Celine's polynomials, Czechoslovak Math. Soc, 49(124) (1999), 527-545.

18

82. KHAN, M.A. On exact D.E. of three variables, Acta Ciencia

Indica, 25M (3) (1999), 281-284.

83. KHAN, M.A. AND : On a new class of polynomials set suggested by

ABUKHAMMASH, G.S. Hermite polynomials, Bulletin of the Greek

84. KHAN, M.A. AND

NAJMI, M.

85. KHAN, M.A. AND

AHMAD, K.

86. KHAN, M.A. AND

NAJMI, M.

87. KHAN, M.A. AND

AHMAD, K.

88. KHAN, M.A. AND NAJMI, M.

Math. Soc, 41(1999), 109-121.

Discrete analogue of Cauchy's integral

formula. Bull. Inst. Pol. Din. lasi., XLV

(XLIX), Fasc. 3-4 (1999), 39-44.

A study of Bessel polynomials suggested by the

polynomials Lf^^ (x) of Prabhakar andRekha,

Souchow J. Math., 26 (1) (2000), 19-27.

Discrete analytic continuation of a (p,q)-

analytic function. Pro Mathematica, 14(27-28)

(2000), 99-120.

On a two variable analogue ofBessels

polynomials. Math. Balkanica (N.S.), 14(27-

28) (2000), 65-76.

A note on a polynomials set generated by G(2-xt-t^)for the choice G(u) = OFl

(-;a;u), Pro Mathematica, 15(29-30) (2001), 5-20.

119

89. KHAN, M.A. AND

KHAN, A.R.

90. KHAN, M.A. AND

NAJMI, M.

On a new class of polynomial sets suggested by

the polynomials Kn(x) of M.A. Khan and G.S.

Abukhammash, Acta Ciencia Indica, 27M(4)

(2001), 429-438.

On a convolution operator for (p,q) analytic

function, Rev. Mat. Estat. Paulo, 19 (2001),

273-283.

91. KHAN, M.A. AND

APIMAD, K.

92. KHAN, M.A. AND

KHAN, A.R.

93. KHAN, M.A. AND

KHAN, A.R.

94. KHAN, M.A.

On certain characterizations and integral

representations ofChatterjea 's generalized

Besselpolynomials, Glasnik Mathematicky,

37(57) (2002), 93-100.

On a triple transformations of certain

hypergeometric functions, Acta Ciecia Indica,

20C3M(3) (2002), 377-388.

A study of a new class of polynomial set

suggested by Gegenbauer polynomials, Math.

Sci. Res. J., 6(9) (2002), 457-472.

On a q-analogue ofEuler 's theorem, Revista

Ciencias Mathematicas, 20(1) (2002), 19-21.

120

95. KHAN, M.A. AND On a generalization ofAppell 's functions of two

ABUKAMMASH, G.S. variables, Pro Mathematica, 16(31-32)

(2002), 61-83.

96. KHAN, M.A. AND : On convex operator for (p,q)-analytic

NAJMJ, M. functions, Novi Sad, J. Math., 32(2) (2002),

13-21.

97. KHAN, M.A. AND A study of q-Lagranges polynomials of three

KHAN, A.R. variables. Acta Ciencia Indica, 29M(3)

(2003), 545-550.

98. KHAN, M.A. AND A study of two variable analogues of certain

ABUKHAMMASH, G.S. fractional integral operators. Pro

Mathematica, 17(33) (2003), 31-48.

99. KHAN, M.A. AND : On a new class of polynomial set suggested by

ABUKHAMMASH, G.S. Legendrepolynomials, Acta Ciencia Indica,

29M(4) (2003), 857-865.

100. KHAN, M.A. AND A study of three variable analogue of

KHAN, A.R. Konhauser 's Biorthogonal polynomial Lf(x;k),

communicated

for publication.

121

lOl.KPiAN, M.A. AN Sister Celine's polynomials of two variables

SHARMA, B.S. suggested by Appell 's function, communicated

for publication.

102. KHAN, M.A. AN A note on a three variables analogue ofBessel

SHARMA, B.S. polynomials, communicated for publication.

103.KHAN, M.A. AN A study of three variables analogues of certain

SHARMA, B.S. fractional integral operators, communicated

for publication.

104. KHAN, M.A. AND On fractional integral operators of three

SHARMA, B.S. variables and integral transforms.

communicated for publication.

105. KHAN, M.A. AND: On a new class of polynomial set suggested by

SINGH, M.P. the polynomial of M.A. Khan and A.R. Khan,

communicated for publication.

106. KHAN, M.A. AND On a general class of polynomials of three

SINGH, M.P. variables L. a P y 51 fj (x,y,z), communicated for

publication.

107. KHAN, M.A. AND: A note on a calculus for the T/, ^ y-operator,

SINGH, M.P. communicated for publication.

122

108. KHAN, M. A. AND : A study of a two variables legendre

SINGH, M.P. polynomials, communicated for publication.

109. KHAN, M. A. SHUKLA, : Solution of mis cible fluid flow through A.K.

AND SINGH, M.P. porous media in terms of hyper geometric

function iFj, communicated for publication.

] 10. KHAN, M.A. AND : On a general class of Polynomial L^^'^'^'^\x,y)

AHMAD, K. of two variables suggested by Re-polynomials

r(a,P) (a,P), D„'^'{x,y) ofRagaband C'^'H^) of

Prabhakaran and Rekha communicated for

publication.

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(1967), 303-314.

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A new class of orthogonal polynomials the

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Some integral equations involving

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