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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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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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Some integral equations involving
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