Rev. Acad. Canar. Cienc., XVII (Núms. 1-2), 37-58 (2005) (publicado en agosto de 2006)
SOME FAMILIES OF MULTILATERAL GENERATING
FUNCTIONS FOR THE GEGENBAUER (OR ULTRASPHERICAL)
POLYNOMIALS AND ASSOCIATED HYPERGEOMETRIC
POLYNOMIALS
Shy-Der Lin
Department of Applied Mathematics
Chung Yuan Christian University
Chung-Li 32023, Taiwan, Republic of China
E-Mail: shyder@cycu.edu.tw
H. M. Srivastava
Department of Mathematics and Statistics
University of Victoria
Victoria, British Columbia VSW 3P4, Canada
E-Mail: harimsri@math.uvic.ca
Pin-Yu Wang
Department of Mechanical Engineering
Nan-Ya lnstitute of Technology
Chung-Li 32023, Taiwan, Republic of China
E-Mail: pinyu@nanya.edu.tw
Abstract
The present investigation is essentially a sequel to severa} earlier works which
appeared recently in this Revista de la Academia Canaria de Ciencias. The authors
begin by showing that a family of bilateral generating functions for the
Gegenbauer (or ultraspherical) polynomials, which was stated incorrectly and
claimed to have been proved by using the familiar group-theoretic (Lie algebraic)
method by A. B. Majumdar [Rev. Acad. Canaria Cienc. 7 {1995), 111-115], is
derivable fairly easily by suitably specializing a known multilateral generating
function. A number of other analogous families of bilateral and multilateral
generating functions for hypergeometric polynomials, including those considered in
many recent papers, are also investigated here.
2000 Mathematics Subject Classification. Primary 33C45, 33C20; Secondary 33C65.
Key Words and Phrases. Generating functions, classical orthogonal polynomials, Jacobi
polynomials, Hermite polynomials, Gegenbauer (or ultraspherical} polynomials, Bessel
polynomials, Laguerre polynomials, Legendre (or spherical) polynomials, Chebyshev polynomials,
bilateral and multilateral generating functions, hypergeometric polynomials, Gauss hypergeometric
function, Group-theoretic (Lie algebraic) method.
37
l. Introduction, Definitions and Preliminaries
It is well known that the classical Gegenbauer (or ultraspherical) polynomials C~(x) are defined
by the following generating function:
00
(1- 2xt + t2rv = ¿ C~(x)tn (1.1)
n=O
or, explicitly, in terms of a hypergeometric polynomial by
-~n, - ~n+ ~;
1-v-n;
1
x2
(1.2)
where 2F1 denotes the familiar (Gauss's) hypergeometric function which corresponds to the special
case
u-l=v=l
of the generalized hypergeometric function uFv with u numerator and v denominator parameters.
These polynomials are a generalization of the Legendre ( or spherical) polynomials Pn ( x) ( for v = ~)
as well as the Chebyshev polynomials Tn(x) (for v -t O) and Un(x) (for v = 1) of the first and
second kinds, and are orthogonal over the interval (-1, 1) with respect to the weight function:
Wv (x) := (1- x2)"-! ( !lt(v) > -~).
More generally, in terms of the classical Jacobi polynomials P~a,P) (x) of degree n in x (and with
parameters or indices a and [3), which are defined usually by
pJa,8) (x) := ~ (:~~) (n;P) ( x; l) n-k (x; l r
= (n: ª) 2F1 (-n,a+ P +n +l;a + l;
1
; x) (1.3)
and are orthogonal over the interval (-1, 1) with respect to the weight function ( cf., e.g., Szego [25,
p. 68, Equation (4.3.3))):
Wa,p (x) := (1 - x)ª (1 + x)P ( min {9l (a), ~t(P)} > -1 ),
that is, L (1 - xt (1 + xJ8 p~a,8) (x) P!a,/l) (x) dx
= 2°+P+Ir(a+n+l)r(,B+n+l) ó (1.4)
n! (a + {3 + 2n + 1) r (a + {3 + n + 1) m,n
( min { 9t (a) , 9t (,8)} > -1; m, n E No := N U {O} ; N : = { 1, 2, 3, ... } ) ,
38
Óm,n being the Kronecker symbol, we have the following special relationship for the Gegenbauer ( or
ultraspherical) polynomials C~(x):
c;:+l (x) = (v: r (2v: n) PJ"•"l(x), {1.5)
Many other members of the family of classical orthogonal polynomials, including (for example)
the Hermite polynomials Hn (x), the Laguerre polynomials L~ª) (x), and the Bessel polynomials
Yn (x; a, /3), are also special or limit cases of the Jacobi polynomials PJa,/J) (x). In particular, for
the classical Laguerre polynomials L~ª) (x) defined by
L~0l (x) := t (:~ :t:i)k = (n: ª) ¡F¡(-n;a+ l;x),
k=O
(1.6)
it is easily observed that [25, p. 103, Equation (5.3.4)]
L(a) (x) = lim {p(a,{J) (i -2x)}
n lfJl-+oo n /3 ' (1.7)
which can indeed be applied to deduce properties and characteristics of the Laguerre polynomials
from those of the Jacobi polynomials.
For the Gegenbauer (or ultraspherical) polynomials C~(x), by applying the familiar grouptheoretic
(Lie algebraic) method, the following family of bilateral generating functions was asserted
erroneously by Majumdar [12, p. 111].
(?) Theorem l. If there exists a unilateral generoting relation of the fonn:
00
G(x,w) = Lan c~-m(x)wn, (1.8)
n=O
then
00
G(x + (x2 - l)wyz,wy) = L c~-m-p(x} O'n(z)(wy}n, (1.9)
n=O
where
O'n (Z) --En -ap- (p-2.X+2m+l}n-p(p+l)n-p Zn -p • p=O(n-p)! 2n-P(-.X+m+l)p
(1.10)
Here, and in what follows in our present investigation, (.X) 11 denotes the Pochhammer symbol (or
the shijted factorial, since (l)n = n! for n E No) defined (for ,\,v E C and in terms of the Gamma
function} by
r ( .x + v) { 1 ( v = o; .x E e \ {o}}
(.X)":= r .X =
( ) .X (.X+ 1) .. ·(.X+ n - 1) (v = n EN:= No \ {O}; .X E C),
(1.11)
A closer examination of (1.8), (1.9) and (1.10) would immediately reveal the fact that the
parameters m and y are redundant (or superfluous), since ( without any loss of generality) we
can trivially replace ,\ by A + m and set y = l. More importantly, the parameter p occurring on
the right-hand side of the assertion (1.9) is conspicuously absent on the left-hand side of (1.9). We
begin this paper by presenting a corrected and modified version of (?) Theorem 1, which is a1so
39
shown here to be derivable fairly easily from one of severa! much more general known families of
multilateral generating functions for the Gegenbauer (or ultraspherical) polynomials C~(x). We
then propase to (systematically) analyze and investigate many other families of (known or new)
families of bilinear, bilateral and mixed multilateral generating functions associated with the
following class of hypergeometric polynomials:
(1.12)
(nENo :={0,1,2, ... }; aEC; vEC\Z0; Zi):={0,-1,-2, ... }),
which have also been considered recently, from a markedly different viewpoint, by ( for example)
Driver and Moller [8], Chen and Srivastava [4], and others (see also the references cited in each of
these earlier works).
2. Generating Functions for the Gegenbauer ( or Ultraspherical) Polynomials
For the Gegenbauer (or ultraspherical) polynomials C~(x) introduced in Section 1, the following
generating functions are known fairly well (ej., e.g., McBride [13, p, 56J):
~ (k - n -2v) C" ( ) k = Rn C" (x -t) f..J k n-k X t n R '
k=O
(2.1)
{2.2)
and
~ (n +k: 2v-1) Gf(x) t" = wn-2v G~ (1 ~xt), (2.3)
where, for convenience, R is given by
1
R := (1 - 2xt + t2)2. {2.4)
As a matter of fact, in view of the hypergeometric representation ( cf. Rainville [18, p. 280,
Equation (20]):
40
v+ !.
2'
(2.5)
or, equivalently,
C11( ) (2v + n - 1) -211-n F
n X = X 2 1 n
which follows from (2.6} by means of Euler's transformation:
v+l· 2'
2F1(a,b;c;z) = (l-z)c-a-b2F1(c-a, c-b; e; z)
(larg(l - z}I ~ 7r - E; o < E < 7rj a, b E C; e E e\ l()),
(2.6)
(2.7)
it is not difficult to show that the generating function (2.3) is actually a special case of the following
well-known result [23, p. 126, Equation 2.4 (9)):
when
~ (,\)k C11(x) tk = (1 - xt)->. F
l..J (2v) k 2 I
k=O k
h l\+l.
2"' 2" 21
v+ !. 2'
,\ = 2v + n ( n E No) .
(x2 - 1) t2
(1 - xt)2
Next, by applying the relationship (1.5) in its relatively more convenient form:
R(11-i,11-i)(x) = (v + ~)n C11(x)
n {2v)n n '
(2.8)
(2.9)
many of the known results involving Jacobi polynomials can easily be specialized to hold true for
the Gegenbauer ( or ultraspherical) polynomials. Thus, for example, we find from the following
known results (see, for example, Srivastava and Manocha [23); see also González et al. [9]):
E ( m: n) P!:';.•,P-n)(x)t• = { 1 + ~(x + l)t}. { 1 + ~(x -l)t r
. P,\,"0Pl (x + ~ (x2 -1) t) (2.10)
t (a+ (3; n +e) p~~;t,P+t)(x)l = p~•,Pl(x + 2t),
t=O
(2.11}
41
~ ().)l p(o-t,{J-l)(x)l
lJ (-a - f3)t t
l=O
,\, -a;
t
1 + ~(x - l)t
(2.12)
-a-{3;
and
(2.13)
that
~ (m+n) (1-2v-m)ncv-n(x)tn
L..J n (1 - v) m+n
n=O n
(2.14)
(2.15)
,\, ~ - v;
4t
1+2(x-l)t'
(2.16)
1-2v;
and
42
respectively.
The finite summation formulas (2.15) and (2.17) are substantially the same result. Indeed, upon
reversing the order of the sum in (2.17), if we replace v and t by v + n and -c1, respectively, we
easily obtain the finite summation formula (2.15). More importantly, since [23, p. 126, Equation
2.4 (6))
C~(x) = (-2y'x2- l )" p~-v-n,-v-n) ( G=-r), (2.18)
which, in view of the relationship (2.9), assumes the following equivalent form:
(2.19)
the generating functions (2.2) and (2.14) are equivalent.
Now, in view of the hypergeometric representation (1.2), by appropriately specializing a known
hypergeometric generating function [9, p. 138, Equation (2.15)], we can deduce the following
generating function (ej. [9, p. 163, Equation {6.18}]):
(2.20)
Indeed, by means of the relationship (2.19), it is not difficult to rewrite this last generating function
(2.20) in its equivalent form:
f (1 - 2v - nhk c11-k(x) ~
k=O {1 - v)k n k!
=(1-4t)"-~{1+4(x2 -l)t}~"c~( x )· {2.21)
J1+4(x2 -l)t
Lastly, if we apply the relationship {2.19), we easily see that the finite summation formula (2.15)
assumes the equivalent form (2.1). Thus the finite summation formulas (2.1), (2.15), and (2.17)
are essentially the same result stated in seemingly different ways.
With a view to obtaining various families of bilinear, bilateral or mixed multilateral generating
functions for the Gegenbauer ( or ultraspherical} polynomials, we first observe that (2.2) leading to
a known result dueto Srivastava [20, Part 1, p. 229, Corollary 4], as well as each of the generating
functions (2.14) [with v replaced trivially by v - m (m E No)], {2.20) [with v replaced trivially by
v + m (m E~)), and (2.21) [with v replaced trivially by v - m (m E No)] fits easily into the
Singhal-Srivastava definition [19, p. 755, Equation (1)):
00 L Am,k Sm+k(x) tk = f (x, t) {g(x, t)}-m Sm (h(x, t)} (m E NQ). (2.22)
k=O
43
Thus, by comparing the Singhal-Srivastava generating function (2.22) with the aforementi.oned
(trivially modified) versions of the generating functions (2.2), (2.14), (2.20), and (2.21), respect1vely,
we obtain the following special ca.ses of (2.22):
A.n,k = (m :k). f = n-2•, g = R, h =X; t. and Sk(x) = Ck(x}; (2.23)
A - (m + k) (1 - 211 + m)k f = {1+4xt + 4 (x2 - 1) t2}v-~'
m,k - k (1 - v + m)k '
g = 1 + 4xt + 4 ( x2 - 1) t2, h = x + 2 ( x2 - 1) t,
and Sk(x) = crk(x); (2.24)
Am,k= (v+m¡k-1), /=(1-tf"-!n, g=l-t,
h = ~' and Sk(x) = c~+k(x); (2.25)
vl-t
A = (1 - 211 + 2m - n)ik f = (1 - 4t)v-~ {1 + 4 (x2 - 1) t} ~n
m,k k!(l - v + m)k ' '
g = 1 - 4t, h = X ' and Sk(x) = c~-k(x). (2.26) J 1 + 4 ( x2 - 1) t
In light of the connections exhibited by (2.23) to (2.26), the entire development stemming from
th~ Singhal-Srivastava generating function {2.22) would readily apply also to each of the generatmg
functions (2.2), (2.14), (2.20), and (2.21). Alternatively, however, by appealing directly to
t~~ gener~ting functions (2.2), (2.14), (2.20), and (2.21), we can derive a set of four families of
hilinear, bilateral or mixed multilateral generating functions for the Gegenbauer (or ultraspherical)
polynomials, which are given by Theorems 3 to 6 below. More importantly, we first make use of
the generating function (2.14) in arder to derive the following corrected and modified version of
the erroneous result asserted by Theorem 1 without applying the group-theoretic (Lie algebraic)
method.
Theorem 2. If there exists a unilateral generating relation of the form:
00
G(x, z) = L lln c~-n(x) zn (an #O), {2.27)
n=O
then
{l + 4xt + 4(x2
- l)t2}•-! G ( x + (x2
- l)t, 1+4xt + ~(x2 - l)t2 )
00
= L c~-n(x) ª"(z) t", (2.28)
n=O
where
~ (n) (1- 211 + k)n-k ~
O"n(z) := Í...J k (1 - 11 + k)n-k ªk .
k=O
(2.29)
44
Proof. In order to give a direct proof of the assertion (2.28) of Theorem 2 without using the
group-theoretic (Lie algebraic) method adopted by Majumdar [12] to prove the obviously incorrect
Theorem 1, we first substitute the definition (2.29) into the right-hand side of (2.28) and invert
the order of the resulting double sum. Then, upon evaluating the inner n-sum by appropriately
applying the generating function (2.14), we simply interpret the remaining k-sum by means of
(2.27).
Much more general families of bilinear, bilateral or mixed multilateral generating functions than
the assertion (2.28) of Theorem 2 are readily accessible in the existing mathematical literature.
We first recall a mild extension of the aforementioned class of bilateral generating functions of
Srivastava (20, Part I, p. 229, Corollary 4] as Theorem 3 below (see also González et al. (9]).
Theorem 3. Corresponding to a non-vanishing function ílμ (y1, ••. , Ys) of s variables
y¡, ... ,ys (s E~
and of ( complex) order μ, let
A~~p,q [x; Yl, ... , Ysi z]
00
:= L an c~1;(x) ílμ+pn (y¡, ... 'Ys) zn
n=O
(an#O; mENo; p,qE~,
where p is a suitable complex parameter. Suppose also that
T~:'!t.v (x; y¡, ... , Ysi z)
[n/q) ( m+n ) 11tpqk
:= [ n-qk akCm+n (x)
k=O
Then
00 L r~:'!t.v (x; y¡, ... ,Ysi z) t"
n=O
-m-211 A(l) [X - t ( t )q] = R m,p,q ---¡¡-; Y1, ... 'Ysi z R2p+l '
provided that each member of (2.32) exists, R being given, as be/ore, with (2.4).
(2.30)
(2.31)
(2.32)
We choose also to recall the following analogous families ofbilinear, bilateral or mixed multilateral
generating functions for the Gegenbauer (or ultraspherical} polynomials (ej. González et al. [9]).
Theorem 4. Under the hypotheses of Theorem 3, let
A~~p,q [x; y¡, ... , Ysi z]
00
:= L ªn c:;;~~(x) ílμ+pn (y¡, ... 'Ys) z"
n=O
45
(2.33)
(an #O; m E No; p,q EN),
where p is a suitable comple:c parameter. Suppose also that
P~:~~P (x; y¡, ... , Ysi z)
[En/q] (m + n) (1- 2v - m + (2p- 1)qk)n-qk cv-n-(p-l)qk( )
~ ~ m~ X
n -qk (1- v + pqk)n-qk k=O
· "u μ+pk ( Y¡, .. · 'Ys ) Z k • {2.34)
Then
00
[ P~:~P (x; Yb .. ·, Ys; z) tn
n=O
1
= { 1 + 4xt + 4 ( x2 - 1) t2 r-2
(2) [ 2 ) • • ztq ] (2 35)
. ~,p,q x + 2 (x - 1 t, y¡, · · ·, Ys, {l + 4xt + 4 (x2 _ 1) t2JPq ' ·
provided that each member of (2.35) exists.
Theorem 5. Under the applicable hypotheses of Theorem 3, let
A~~i,q [x; y¡, ... , Ysi z]
00 :=E ªk c~+(p+I)qk(x) ílμ+pk(y¡, ... 'Ys) zk
k=O
(ak ;t O; n E No; p, q EN),
where p is a suitable comple:c parameter. Suppose also that
Then
00
Q~;;:: (x; y¡, ... , Ysi z)
[k/q]
·= "'"" (V + k + pqi - 1) cv+k+pqi( )
• l...J k - i al n X
l=O q
· ílμ+pt (y¡, . ·., Ys) /·.
[ Q~;;:: (x; y¡, ... , Ysi z) tk
k=O
(2.36)
{2.37)
( -11-ln A(3) [ x . . ztq ] (2 38)
= 1 - t) 2 n,p,q v'l _ t 1 y¡, .. · 1 Ys1 (l _ t)(p+l)q ' ·
provided that each member of (2.38) e:cists.
46
Theorem 6. Under the applicable hypotheses o/ Theorem 3, let
A~~,q [x; y¡, ... , y8 ; z]
00 k
:= L ªk c~-pqk(x) ílμ+pk(y¡, ... 'Ys) ( zk)'
k=O q '
(ak7~ O; n E No; p,q EN),
where p is a suitable complex parameter. Suppose also that
,,,11,μ,p (x· y y . ) l\.k,p,q , ¡, ... ' 87 z
["k/q') (k) (1- 2v + 2pql- n)2(k-qt) k ( l) l := l..J aec~- - p- q (x)
l=O ql (1 - V+ pql)k-ql
• ílμ+pt( y¡, .. ·, Ys) zt.
Then
00 tk
"l.'.,J,, 11,μ,p (x· y . ) l\.k,p,q ' ¡, ... , Ys, z kI
k=O
1 1
=(1-4t)11-2 {1+4(x2 -l) t}2"
. Al.~,q [ y'l +4 ~xL 1) ti y¡, ... 'y,¡ (1 ~t)pq] '
provided that each member of (2.41) exists.
(2.39)
(2.40)
(2.41)
We now turn once again to the known finite summation formulas (2.1), (2.15), and {2.17).
Indeed, as we have already indicated above, the finite summation formulas (2.15) and {2.17) are
substantially the same result: One follows from the other by merely reversing the order of terms
in the finite sum and making sorne obvious variable and parameter changes. In order to show
the equivalence of the finite summation formula (2.1) with (2.15) or (2.17), we simply apply the
relationship (2.19) appropriately. Thus the three (seemingly different) finite summation formulas
(2.1), (2.15), and (2.17) are equivalent to one another.
Although any of the (already proven equivalent) finite summation formulas (2.1), (2.15), and
(2.17) does not really fit into the Singhal-Srivastava definition (2.22), yet Uust for the sake of
completeness) Theorem 7, Theorem 8, and Theorem 9 below can be shown to follow analogously
from the finite summation formulas (2.1), (2.15), and (2.17), respectively.
Theorem 7. Under the applicable hypotheses o/ Theorem 3, let
A~1,q [x; YI, · . · , Ys; z)
[n/q]
:= L ªk c~~~k(x) ílμ+pk (y¡, ... , Ys) zk
k=O
47
(2.42)
where p is a suitable complex parameter. Suppose also that
u:::,: (x; y¡, ... , Ysi z)
[k/q] f.) == L (k- n - 2v ¡ 2pq ªt c:~~ql(x)
l=O k-q
· ílμ+pt( y¡, ... , Ys) i.
Then
n
"u;,μ,p (x; y¡, ... , Ys; z) tf L., ,p,q
k=O
[
= R" ~51,q RX - ;t yi, . .. ,Ysi ( t )q] Z R '
where R is given (as be/ore) with (2.4).
Theorem 8. Under the applicable hypotheses o/ Theorem 3, let
~~i,q [x; y¡, ... , Ys; z]
[n/q] k
:= L akc:~;t(x)ílμ+pdY1, ... ,ys) z
k=O
( ak # O; n E No; p, q E N) '
where P is a suitable complex parameter. Suppose also that
v;:,: (x; y¡, ... , Ysi z)
·- [Lk/qJ (v + k + (p- l)qf. - 1) a cv+k+(p-I)qt(x)
.- 0 l n-k k-qc.
l=O
· ílμ+pt( y¡, .. ·, Ys) zt.
Then
n
"' Vkv,μ,p (x; y¡, ... , Ysi z) tk
'--' ,p,q
k=O
= A(6) [x+ !t; y¡, ... ,ys; ztq] · n,p,q 2
Theorem 9. Under the applicable hypotheses o/ Theorem 3, let
A~:i,q [x; y¡, ... , Ysi z]
(2.43)
(2.44)
(2.45)
(2.46)
(2.47)
[~] (n + pqk - v)-l v-n-pqk( ) " ( ) k
:== ¿_, n _ qk ak Cn-qk X uμ+pk Yl, · · ·, Ys Z {2.48)
k=O
(ak?f O; n E No; p,q EN),
48
where p is a suitable complex parameter. Suppose also that
wz:;,: (x; y¡, ... , Ys; z)
lk/q] ( )( )-1 := ~ n - ql k + pql - V a cv-k-pql(x)
í..J n - k k - ql l k-ql
l=O
Then
n L wz:;,: (x; y¡, ... ,ys; z) tk
k=O
= t• A~~.q [x - ;t; y¡, ... ,y,; z] .
(2.49)
(2.50)
Remark l. The special case of Theorem 3 when p = O is precisely the aforementioned class of
bilateral generating functions of Srivastava [20, Part I, p. 229, Corollary 4] (see also [23, p. 422,
Corollary 4]). This result of Srivastava [20, Part I, p. 229, Corollary 4] can indeed be shown to
yield, as its obvious further special cases, numerous families of bilateral generating functions, which
are scattered all over the subsequent mathematical investigations (see, for details, González et al.
[9, p. 166 et seq.]).
Remark 2. Various known special cases of sorne of the other results of this section include
special cases of Theorems 4, 5, and 6 when (for example)
p = q = 1, p =O, and ílμ (yi, ... , Ys) = 1 (2.51)
or
p=q=p=l and ílμ (y¡, ... ,y8 ) = 1 (2.52)
or
q=p=l and ílμ (y¡, ... ,y8 } = 1 (2.53)
or
q = 1, p=O, and ílμ (y¡, ... ,y8 } = 1 (2.54)
were also rederived in many recent investigations (see, for details, González et al. [9, p. 166 et seq.]).
If, in the special case listed under (2.52), we further set m =O, Theorem 4 would immediately yield
Theorem 2, that is, the duly-corrected version of the emmeous result asserted by Theorem l.
Remark 3. Since the generating functions (2.2) and (2.14} are equivalent, as we observed above
by using the relationship (2.19), their consequences (Theorem 3 and Theorem 4) are also equivalent.
Furthermore, since {2.14) itself is a special case of (2.12), Theorem 4 {and hence also Theorem 3)
can alternatively be deduced from a result of Srivastava and Popov [24, p. 178, Theorem) by setting
p = a (see also Srivastava and Handa [22] for further extensions involving a general sequence of
functions defined by a Rodrigues formula).
Interconnections of sorne of the other results of this section can also be established by merely
examining the generating functions or the finite summation formulas which actually lead us to
these results (see, for details, González et al. [9, p. 167 et seq.]).
49
3. Generating Functions for Hypergeometric Polynomials
In a recent paper published in this Revista, Mukherjee [14] considered the following very special
case of the Gauss hypergeometric polynomial defined by (1.12):
~ (-Nh (n)k xk
2F1 (-N,n;v+n;x) = ~ (v +n) -k, k . k=O
(3.1)
(n, NE No; 11 E C \ Z0),
where, as usual, ~ and Z0 denote the sets of nonnegative and nonpositive integers, respectively.
By suitably interpreting the parameter n, instead of N (which is done generally [23, Chapter 6,
Section 6.7]), Mukherjee [14] derived several bilateral and mixed trilateral generating relations
for the special Gauss hypergeometric polynomial (3.1) from the group-theoretic (Lie algebraic)
viewpoint.
The main results of Mukherjee [14] are being recalled here as Theorem 9 and Theorem 10 below.
Theorem 9 (e/. [14, p. 123, Theorem l]). lf there exists a generating relation of the form:
then
where
00
G(x,z) = Lªn2F1 (-N,n;v+n;x)zn,
n=O
( )
00 x+t zt n G -
1
- ,- = ~ 2F1(-N,n;11+n;x)pn(z)t,
+t l+t ~
(z)·-~(-lt-k (k)n_dv +N +k)n-k k
Pn .- ~ ( _ k)' ( k) ak z .
k=O n . v + n-k
(3.2)
(3.3)
(3.4)
Theorem 10 (ej. [14, p. 124, Theorem 2]). If there exists a generating relation of the form:
00
H (x,y,z) = L an 2F1 (-N, n; 11 + n; x) 9n (y) zn, (3.5)
n=O
then
(
X+ t zt ) ~ n H lt,y,- 1
- = ~ 2Fi(-N,n; 11+n;x)an(Y,z)t ,
+ + t n=O
(3.6)
where
(3.7)
Remark 4. Obviously, sin ce the (identically nonvanishing) coefficients an ( n E No) are
essentially arbitrary, Theorem 10 can, in fact, be deduced from Theorem 9 itself by appropriately
letting
(n E No) (3.8)
50
in (3.2) and (3.4). Theorem 9, on the other hand, follows from Theorem 10 when we set
or when we let
9n (y)= 1
lln i-> -ª"-
Yn (y)
(n E No)
(n E~).
(3.9)
(3.10)
However, with a view to applying such results as (3.6) above in order to derive bilineax and bilateral
generating relations for simpler special functions and polynomials, it is usually found to be
convenient to specialize an and Yn (y) individually as well as separately ( and in a manner dictated
by the problem). Thus, for the salce of such conveniences in applying these results, we choose to
state the following unification and generalization of Theorem 9 and Theorem 10 in the form of
the mixed multilateral generating relations given by Srivastava [21, p. 3, Theorem 3) (see aiso
Theorems 3 to 8 of Section 2).
Theorem 11. Corresponding to an identically nonvanishing function ílμ (y1, •.. , Ys) of s (real
or complex) variables Y1, ... , Ys ( s E N) and of ( complex) order μ, let
00
Aμ,p [x; y¡, ... , Ysi z] := L l1n 2F1 (A, 'Y+ n; v + n; x)
n=O
· ílμ+pn (y¡, ... , Ys) z"
(ak #O; k E~; p EN:=~ \ {O}),
where .X and 'Y are suitable complex parameters. Suppose also that ·
~ (-1)"-k ('Y+ k}n-k (v - A+ k)n-k
<Pn,μ,p (y¡, ... ' Ysi z} := Í-1 (n - k)! (v + k)
k=O n-k
(3.11)
· ak ílμ+pk(y¡, ... , Ys) i. (3.12)
Then
00 L 2F1 (.X, 'Y+ n; v + n; x) <Pn,μ,p (yi, ... , y8 ; z) t"
n=O
= (l + t¡-1 Aμ,p [~::;y¡, ... , y,; l ~ t] (ltl < 1), {3.13)
provided that each member of (3.13) exists.
Proof. A direct proof of the general result (3.13) is based upon the following well-known
bypergeometric reduction formula [1, p. 24, Equation (28)] (see also [23, p. 105, Equation 2.3
(6)]):
Fi(a, ¡3,,B'; ¡3 +P'; x, y] = (l - y¡-Q 2F1 ( a,¡3;¡3+ /l'; ~ =:) (3.14)
(IYI < 1; a, {3, {3' E C; f3 + {3' E e\ Z()),
where Fi denotes one of the four Appell functions defined by [1, p. 14, Equation (11)]
51
F [ R {3'· . ] ·= ~ (a)m+n (f3)m (f3')n Xm Yn
1 a,JJ, ,-y,x,y . L._¡ ( ) 'Y m+n m.1 n.1 m,n=O
(3.15)
(max{lxl' IYI} < 1; a,{3,{3' E C; 'Y E e\ Z()).
The details involved are being omitted here ( cf. [21, pp. 4-5]).
Remark 5. It is easily observed that, in its special case when
A=-N (NENo), -y=O, and p = s = 1 (y1 =y)'
Theorem 11 would correspond essentially to Theorem 10. And, just as we pointed out in Remark
4 above, Theorem 9 differs from Theorem 10 only notationally in view of (3.8) and (3.10).
Next, by suitably interpreting the parameter a, instead of n (which is done usually (see [23,
Chapter 6, Section 6.7]), Mukherjee [16] derived a family of mixed trilateral generating relations
for the Gauss hypergeometric polynomial (1) from the viewpoint of Lie groups and Lie algebras.
We begin by recalling here the main result of Mukherjee [16] in the following ( slightly modified)
form:
Theorem 12 (e/. [16, p. 57, Theorem]). If there exists a generating relation of the following
/orm:
00
G (x,y;z) = Lan2F1 (-n - r, a; v;x) Un (y) zn, (3.16)
n=O
then
00
= LPn (x,y;z) tn, (3.16)
n=O
where
~ (1-v)n-k k
Pn(x,y;z} := L..Jªk (n-k)! 2Fi(-n-r,a;v-n+k;x)gk(y)z.
k=O
(3.17)
In an earlier work on this same subject, Chongdar and Pan [6] derived the following equivalent
version of Theorem 12 when (see also Remark 4 above)
Un (y) = 1 (n E No). (3.18)
Theorem 13 (e/. [6, p. 217, Theorem 8]). If there exists a generating relation of the form:
00
F(x,z) = Lan2F1 (-n- r,a;v;x)zn,
n=O
52
(3.19)
then
where
(1-w-1 (1 - xtrº F (X (l - t)' zt)
1-xt
00
= :~::>n (x, z) tn,
n=O
~ (1-v)n-k k
un(x,z) := L.,,ªk ( -k)' 2Fi(-n-r,a;v-n+k;x)z.
k=O n .
(3.20)
(3.21)
Remark 6. J ust as we observed above about the equivalence of Theorem 9 and Theorem 10
(see Remark 4), Theorem 12 and Theorem 13 are also equivalent.
Remark 7. An obviously efToneous version of a special case of Theorem 12 when r = O was
claimed to be a new result in a very recent paper by Chongdar and Sen [7, p. 85, Corollary 7].
An interesting unification ( and generalization) of the equivalent results (Theorem 12 and Theorem
13) is provided by Theorem 14 below (ej. Lin et al. [11, p. 611, Theorem 3)).
Theorem 14. Corresponding to an identically nonvanishing function ílμ (y¡, ... , Ys) of s (real
or complex) variables y¡, ... , Ys ( s E N) and of ( complex) order μ, let
00
AW,q) [x; y¡, ... , Ys; z] := L an 2F1 (p - qn, a; 1 - A; x)
n=O
· ílμ+pn (y¡,···, Ys) Zn
( ak # O; k E No; p, q E N; A E C \ N) ,
where p and a are suitable complex parameters. Suppose also that
Then
[n/q) (.A)n- k
wn(x;y¡, ... ,y5;z):= Lak (n-q~)! 2Fi(p-n,a;l-A-n+qk;x)
k=O
00 L wn (x; yi, ... , Ysi z) tn = (1 - tr.\ (1 - xtrº
n=O
A (p,q) ¡x (1 - t). . ql
• μ 1 - xt 'Yl' ... 'Ys, zt
(ltl < min { 1, lxl-1
}) ,
provided that each member of (3.25) exists.
(3.23)
(3.24)
(3.25)
Proof. In the direct proof of Theorem 14, without using the group-theoretic (Lie algebraic)
technique employed by Mukherjee (16] in his rederivation of the very specialized cases given by
53
Theorem 12 and Theorem 13 above, one would require the following hypergeometric generating
function [11, p. 612, Equation (18)]:
00 (,\) L --f1 2F1 (p - n, a; 1 - ,\ - n; x) tn
n=O n.
= (1 - t¡->. (1 - xt¡-ª 2F1 (p, a; 1 - >.; xl(~ ~:)) (3.26)
(!ti< min { 1, lxl-1
}; p, a E C; ,\E C \ N) ,
which Lin et al. [11] derived as one of the two special cases of a general family of hypergeometric
generating functions proven by them [11, p. 612, Equation (16)]. The details involved may be
omitted here.
Remark 8. It is easily seen that, in its special case when
p=-r (rENo), .\=1-v, and p=q=s=l (y¡=y),
Theorem 14 would reduce immediately to Theorem 12. And, justas we pointed out in Remark 8,
Theorem 13 differs from Theorem 12 only notationally in view of (3.9) and (3.10).
In the theory of generating functions (see, for details, [23]), various families of generalized
hypergeometric polynomials in one, two and more variables have also been investigated (see, for
example, [2], [3] and [10]). In particular, we recall here the following well-known (rather classical)
result [23, p. 138, Equation 2.6 (8)]:
L-, u+1Fv X t
. oo (,\)n [-n,a¡, ... ,au; ] n
n=O n. /3¡, ... ,f3vi
xt ]
1-t
(3.27)
(ltl < 1; .\,aj E C (j = 1, ... ,u); f3j E C \ Z0 (j = 1, ... ,v) ).
Now, since [23, p. 42, Equation 1.4 (3)]
[
-n, a¡, ... , au; ] ( ) ... ( )
F: _ a¡ n au n ( )n
u+l v . Z - (f3i)n ... (f3v)n -z
/3¡, ... 'f3v,
[
-n, 1 - /31 - n, ... , 1 - f3v - n; {-l)u+v]
· v+lFu z ' 1 - a1 - n, ... , 1 - au - n;
(3.28)
54
where we have simply reversed the order of the terms of the generalized hypergeometric
polynomials occurring on the left-hand side, it is fairly straightforward to rewrite the well-known
(rather classical) result (3.27) in the following equivalent form:
f, {.\)n (a¡)n ... (au)n [-n, 1 - /31 - n, ... '1 - f3v - n; ] tn
l...J ) ... ( ) v+tFu X ni
n=O (/31 n f3v n 1 - a¡ - n, ... , 1 - au - n; ·
{3.29)
which follows also by suitably specializing some general families of generating functions derived
during the 1960s and 1970s by Srivastava (ej., e. g., (23, p. 145, Equation 2.6 (30)], it being tacitly
assumed that both sides of (3.29) exist.
In a very specialized case of this last generating function (3.29) when
u= v = 1 (a1 = 1- p; /31 = 1 - a= .X),
we easily find for the Gauss hypergeometric polynomials that [e/. Equation (3.26))
f (l -t)n 2F1 (-n, a - n; p- n; x) tn = (1 + xt)ª-P[l - (1- x)t]p-l. (3.30)
n=O n.
Two equivalent results involving the Gauss hypergeometric polynomials occurring in (3.30) were
asserted incomctly by Mukherjee [15, p. 255, Theorems 1 and 2), each of which do not seem to
make any sense whatsoever. Here we make use of the substantially more general result (3.29) with
a view to deriving the following companion of { for example) Theorem 14.
Theorem 15. Under the hypotheses of Theorem 14, let
00
BW,q) [x; Y1, ... , Ysi z] := L an ílμ+pn (y1, · · ·, Ys)
n=O
[
.X + qn, a¡ + qn, ... , au + qn; ]
· u+1Fv X zn
/31 + qn, ... 'f3v + qn;
{3.31)
SS
where .X, a; (j = 1, ... , u), and f3; (j = 1, ... , v) are suitably constrained complex parameters.
Suppose also that
(n/q]
~ (x; Y1, • · ·, Ysi z) := Í: ak ílμ+pk(y¡, · · ·, Ys)
k=O
Then
00
(A+ qk)n-qk (a1 + qk)n-qk · · · (au + qk)n-qk
(n - qk)! (/31 + qk)n-qk · · · (f3v + qk)n-qk
[
-n + qk, l - {31 - n, ... , 1 - f3v - n; ] k
. v+1Fu X Z •
1 - a1 - n, . .. , 1 - au - n;
L Kn (x;y¡, ... 'Ysi z) tn = [1 + (-1r+v xtr"
n=O
• o;:;-(p,q) [ t . Yi y . ___zt _q-~]
'-'μ 1 + (-l)u+v xt' ' .. ·' 8
' [1 + (-l)u+v xt]q
provided that each member of (3.33) exista.
(3.32)
(3.33)
We conclude our present investigation by remarking that, in several recent works (see, for
example, [5] and [7]), the existence of sorne generating functions for the product of two Gauss
hypergeometric polynomials was claimed. In particular, the following bilinear sum was in volved in
Chongdar's work (5, p. 120, Theorem 7]:
00
Í: an 2F1 ( -n, /3; v; x) 2F1 ( -n, {3; n; x )wn,
n=O
which is obviously meaningless, since the denominator parameter n of the second hypergeometric ·
polynomial ought to be restricted by [e/. Equation (1.12)]
n ~ Z0 :={O, -1, -2, ... }.
56
Acknowledgements
The present investigation was supported, in part, by the National Science Council of the
Republic of China under Grant NSC-94-2115-M-033-004, the Faculty Research Program of Chung
Yuan Christian University under Grant CYCU-94RD-RA001-10108, and the Natural Sciences and
Engineering Research Gouncil o/ Ganada under Grant OGP0007353.
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