Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3, August 1995, pp. 297 301. Printed in India.

Certain bilateral generating relations for generalized hypergeometric functions

MAYA LAHIRI and BAVANARI SATYANARAYANA Section of Mathematics, Mahila Mahavidyalaya, Banaras Hindu University, Varanasi 221005, India

MS received 30 August 1994; revised 27 January 1995

Abstract. Recently, we introduced a class of generalized hypergeometric functions I~i~I (x, w) f(x + w) - f(x) by using a difference operator A ..... where A~,~f(x) . In this paper an w attempt has been made to obtain some bilateral generating relations associated with l',(x, w). Each result is followed by its applications to the classical orthogonal polynomials.

Keywnrds. Generalized hypergeometric functions; difference operator; bilateral generating relations; classical orthogonal polynomials.

1. Introduction

In the previous paper [2] we introduced a class of generalized hypergeometric functions ~n;lbq)map)t~ ~ ,,,~"" ! defined by using a difference operator as follows:

m"p)tx w 1 A" (x - w)tt~+"~w] x (ap), -- W; (bq); w -,~tb~), , ) = n !(x - w)t'~] x,w I p + 1Fq ( )l , (1.1) where p+~Fq denotes the generalized hypergeometric functions (see, for example, Srivastava and Manocha [8]). We also derived the following relation:

-n,-; --; (1 I wx wx ] l~(aP)(v uf~ ~__. "~- 00n b"p:2;1 (ap): ";tb~)'~'-/ n[ -q:,;0 (b~): 1+~; -; w,w , (112) where Fq:s;p:r;uv(x, y) is a double hypergeometric function. (see Srivastava and Karlsson [7, p. 27(28)]). The following definitions and results given by Konhauser [1, p. 303(3)], Srivastava and Manocha [8, p. 243(11)] and Manocha [4, p. 687(1.3)] have been used here in regard to the bilateral generating relations for the generalized hypergeometric function U,(x, w):

Z:(x,k) = (, 1Fk_--( n;A(k; 1 + ~); ~ , (1.3) n. k where ~+1 ct+2 ~t+k A(k;l+ct)- k ' k ..... k (k=1,2,3...);

(1 + ~t)m(1 + ct + m), eXlFt (ct + n + m + 1; 1 + 0t; - x); L~)+"(x) = (n + m)! (1.4)

297 298 Maya Lahiri and Bavanari Satyanarayana

.=o ( - o~ - fl + m ). - m + " ,o , .

- (l+~+fl-m)m 1+~ l+x)t m!

x f I -fl;2+m,-m; -e-fl;l + +x)t' 1 +x ' (1.5) where F 1 is an Appell function [6]. We also derived the extended linear generating relation [3] as follows:

~ (n+m~ (2). '~;'"p' '~w)t" .=o \ n 1(1 + ~t + m).'"+":~b'~'"'

(ap)::x; : 2; m; x. ] [ w wt l_t,w, w , . . . . (1.6) where F (3) is Srivastava's general triple hypergeometric series (see, e.g., Srivastava and Manocha [8, p. 69(39)-1).

2. Bilateral generating relations We have derived the following bilateral generating relations for the generalized hypergeometric function P.(x, w):

S"~ n!(q). Z~(y,k)l~,(x,w)t" .~o (1 + ct).(1 + fl)k.

=(1 -- r~l- rt ~E'P + 2:0'0'0' 1 [q: 1, 1, 1,0], [(ap): 1,0, 1, 1], ~ --q+k+ 1:o,o,o,o [1 + ~: 1,0, 1,0], [(bq): 1,0, 1, 1], ; ;[ wt • -- --, h, - wh, w ' (2.1) [A(k:l +fl):O, 1, 1,0]:-; -; -; -; 1-t where F is a generalized Lauricella hypergeometric function of 4 variables and h=

Proof From (2.1), we have n!(q). Z~,(y, k) l~,(x, w) t n .@o (1 + ct).(1 + fl)k. Generalized hypergeometric functions 299

/y'~U

n=O /=0 (1 + n)l '

(,7), ~ (n + l~ (,1 + l). gX z:o .:o. n J(l+~+/)..+ll,w)t"

+2.oool I [~: 1, 1, 1,0], [(ap): 1,0, 1, 1], = (1 - t)-~F~§ 1:6.6,o.o / [1 + e: 1, O, 1,0], [(b,): 17 O, 1, 1],

k

x 1-t 'h'-wh'w [A(k:l+fl):0,1,1,0]: ; ; ; ; [using (1.5)]. This completes the proof of (2.1).

Applications (i) By setting p = q and as = bs(j = 1, 2 .... p) in (2.1), we get

n!(rl). Z~(y,k)J~(x,w)t" n=O"-" (1 + ~t).(1 + fl)k.

[r/:: ; ; x ; wt = (1 -- t)-~F ~3) Y 1 --t 'h' -wh , .... ;A(k;l+fl);l+~: ; {2.2) where h = ~ ~ and J~(x, w) is a modified Jacobi polynomial studied by

Parihar and Patel [5]. (ii) On taking k = 1, p = q, (as) = (bj) and letting w ~0 in (2.1), we get the known result given by Srivastava and Manocha [8, p. 133(9)]. The following results can also be deduced by using the same technique as followed in the previous result.

(m + n)! L~+.(x)i~(y ' w)tn = (1 + ~t)meX(1 - t)- 1 .=o (1 + fl).

-- :: - ; l + ct + m;(a,): Y" " Y " 1 W' ' W' X wt W, t' -- xF(a) "" , -;(bq): -,'l+ct;l+fl; 1- 1-t

[using (1.4)]. (2.3) 300 Maya Lahiri and Bavanari Satyanarayana

Applications (i) By writing p = q and (ai) = (bj) in (2.3), we have (m + n)! L~+.(x)J~(y ' w)t" .:o (1 + fl). = (1 + ~),.e~(1 - t)- 1 -,-,, ( w, x) Wt 1 + ct +m'--Y; 1 +fl; 1 + ~t; --t' (2.4) w 1 1 t where W~ is Humbert's function defined in [7, p. 26(21)] and J~(x,w) is a modified Jacobi polynomial studied by Parihar and Patel [5]. (ii) Taking limit as w --* 0 in (2.4), we obtain the result given by Srivastava and Manocha [8, p. 160(70)].

n~ p{a-n,p-n~lv]l~'ly w)t n

n=O (_~-fl).

= hl+yl~p +3:0,1.0,0 -q + 2:0,0,0,0 [(bq): 1, 1,0, 1], [1 + 7:1,0,0, 1],

[--fl:0,0,1,1]:--; --w:l ; ; (l + y)wth • _. 2 w, th; - wth [-a-fl:O,O,l,1]: ; ; ; 1 [using (1.5)]. (2.5) where h = {1 + + 1)t}- '. Applications The following applications are obvious: n~ P"-",~-"",,~Jrtx w)t" .=o(- ~-fl).-" '" "' ' x l+7::-;--fl; : ; ; ; w = h i + ~Ft3) (y + |)wth th, - wth ] :: ; a-fl;l+7: ; ; 2 ' (2.6) whereh= {1 + 1)t}-x | n! Pt.~-"'~-"~(y) L.(x)t" .~o(-~-fl).

=hl+~FO~[ l+7::-;-fl; ; ; ; (y + 1)xth, th, - xth ]

:: ; ~-fl;l+7:-;- 5 - 9 2 (2.7) Generalized hypergeometric functions 301 where h = {1 + + 1)t} - x

.=o(~ (~).(6).l+~)l.(x, ~ w)F4(y+n,f+n;pl,p2;Z1,Z2)t .

---~ b.,p+ 2:0,0,0,1,1 [-]1:1, 1, 1, 1,0],(6:1, 1, 1, 1,0], [(ap):0,0,0, 1, 1]: -q:l,l,O,l,O [(bq):O,O,O,l, 1]:pl;P2; _;

x x ______. , , ~ , , w w Z1,Zz,t ,-wt,w (2.8) 1 +e; -; -;

Applications [As usual, we get] ~ (~).(~) .... ------a.~x,w)F4( 7 + n, 6 + n;pl,P2;Z1,Z2)t n .=o(1 +~).

x = F~)(y,6; , , , w;px,p2, -, 1 + ct;Z1,Z2,t, --wt), (2.9) where Ua") is a Lauricella hypergeometric function of n variables (see l-8, p. 60(1)]).

- (y).()6 .... ~, : ~ L.(x)r4ff + n,6 + n;pl,p2;Z1,Zz)t ~ n=O I I -I- OC)n =F~4)(Y, 6: , , , ;Pl,P2, -,1 @~t, Zl,Z2,t , -xt). (2.10)

Acknowledgements The authors take this opportunity to express their sincere thanks to Prof. H M Srivas- tava (University of Victoria, Canada) for his valuable suggestions and helpful criticism in the preparation of this paper.

References

[1] Konhauser, Joseph D E, Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 21 (1967) 303-314 [2] Lahifi M and Satyanarayana B, A class of generalized hypcrgeometric functions defined by using a difference operator, Soochow J. Math. 19 (1993) 163-171 [3] Lahiri M and Satyanarayana B, Extended linear and bilinear generating relations for a class of generalized hypergeometric functions, Indian J. Pure Appl. Math. 24 (1993) 705-710 [4] Manocha H L, Bilinear and trilincar generating functions for Jacobi polynomials, Proc. Cambridge Philos. Soc. 64 (1968) 687-690 [5] Parihar C L and Patel V C, On modified Jacobi polynomials, J. Indian Acad. Math. 1 (1979) 41-46 [6] Rainville E D, Special functions, (New York: Macmillan)(1960) [7] Srivastava H M and Karlsson P W, Multiple Gaussian hypergeometric series, (Ellis Horwood Limited, Chichestcr; Halsted Press) (New York: John Wiley) (1985) [8] Srivastava H M and Manocha H L, A treatise on generating functions, (New York: Halsted Press, John Wiley) (1984)