Some extensions of Bateman's product formulas for the Jacobi polynomials

Journal of Applied Mathematics and Stochastic Analysis 8, Number 4, 1995, 423-428 SOME EXTENSIONS OF BATEMAN’S PRODUCT FORMULAS FOR THE JACOBI POLYNOMIALS MING-PO CHEN Institute of Mathematics A cademia Sinica Nankang, Taipei 11529 Taiwan E-mail: H.M. SRIVASTAVA University of Victoria Department of Mathematics and Statistics Victoria, British Columbia V8W 3P Canada E-mail: (Received October, 1994; Revised May, 1995) ABSTRACT The authors derive generalizations of some remarkable product formulas of Harry Bateman (1882-1946) for the classical Jacobi polynomials. They also show how the results considered here would lead to various families of linear, bilinear, and bilateral generating functions for the J acobi and related polynomials. Key words: Product Formulas, J acobi Polynomials, Generating Functions, Series Inversion, Linearization Formula, Gaussian Hypergeometric Function, Hypergeometric Identity, Quadratic Transformation, Appell Functions, Polynomial Expansions, Complex Sequence, Dixon’s Summation Theorem, Clausenian Hypergeometric Series, Polynomial Identity, Reduction Formula. AMS (MOS) subject classifications: 33C45, 33C65, 33C20. 1. Introduction and Preliminaries . As long ago as 1905, Bateman [6] gave the remarkable product formula: (a+Z+2k+llk!r(a+Z+k+l) ( )r( + + + + ) o (1) F(a + n + 1)F( +n + P’)(z)Pa’Z)(Y)’ r(a + k + 1)F(Z + k + 1 from which, by applying an elementary series inversion [13, p. 388, Problem 74], it is not difficult to deduce the following linearization formula for the classical Jacobi polynomials (-1) n+ k(a + fl +n + 1)k[x + y + k P(na’ x )P(na’ ) y) 2 n!(n k)! (2) 11 k=OE Printed in the U.S.A. ()1995 by North Atlantic Science Publishing Company , 423 MING-PO CHEN and H.M. SRIVASTAVA 424 r(a + n + 1)r(Z + n + 1)p(C,, )f i + xy F(c + k + I)F( + k + 1) k which was indeed proved directly by Bateman [7, p. 392] by showing that both sides of (2) satisfy the same partial differential equation. Here, and in what follows, (),: F(A + #)/F(A), in terms - P(n’’)(x) denotes the Jacobi polynomial of degree n in x, defined by of Gamma functions, and (cf., eg., Szeg5 [13, Chapter 4]) P(na’)(x)" --k_o( n+a n k (n+)k (x !)k(x+12 (3) Each of Bateman’s formulas (1) and (2) has been applied in the literature in a number of different directions (see, for details, Askey [2, pp. 11 and 33]). In addition, Bateman’s formula (1) was applied by A1-Salam [1] in order to derive the following interesting result due to Feldheim n-o n!(c +/ + 1)n(7)n(5)n (c + 1),(/ + 1)n(a +/ + - - 1)2,P(’Z)(x)P(’)(Y) 2F1(7 + n, 5 n; c -I-/ -+- 2n (4) 2; t)t n, where 2F1 is the Gaussian hypergeometric function and F 4 denotes one of Appell’s double hypergeometric functions defined by xp yq E (a)p+q(b)p+q (C)p(d)q p! q! F4[a’b;c’d;x’Y]" p,q (5) 0 1 1 (112+ lyl2<1), It should be noticed in passing that, in the particular case when 7 1/2(a + + 1) and 5 1/2(a +/ + 2), Feldheim’s formula (4) would reduce to Bailey’s bilinear generating function for the Jacobi polynomials [5]: n__o (1 + n!(a + + 1)n p(na, )(x)p(n )(y)t n ’ t)c--lrr[1/2(o + -" 1),1/2(a + + 2);a + 1,/ + 1; X, Y] X’-(1-x)(1-y)t" y: (l+t)2 (6) (l + x)(l + y)t (l+t)2 ]’ in view, of course, of the familiar hypergeometric identity (cf., eg., Erdlyi et al. [8, p. 101]): 2 2Fl(a1/2,a;2a;zl ( l+v/i-z )2a-1 (Izl < 1), (7) which is, in fact, a special case of the following quadratic transformation for the Gaussian hypergeometric function [8, p. 111, Equation 2.11(10)]: Some Extensions of Bateman’s Product Formulas for the Jacobi Polynomials 425 2F1 2a, 2b;a + b -}- (8) 2Fl(a,b;a -[- b q- ;z) (I z < 1) 2 1-1/2. For several further applications of (1) in the theory of generating functions, one when bto a recent treatise on the subject by Srivastava and Manocha [12, Chapter 2, Problem may refer 14]. Motivated by the aforementioned potential for applications of Bateman’s formulas (1) and (2), we aim here at investigating some interesting generalizations of (1). We also show how these general results can be applied in the theory of generating functions. 2. Polynomial Expansions in Several Variables We begin by introducing the class of multivariable polynomials 0 defined by M<n E k1 (M" mlk I + + rarer; mj E I: kr 0 n)M(’ q- n)MA(kl’"" "’ kr)Zlkl’" "Zrr {1,2,3,...} (j 1,...,r); e C\{0, 1, (9) 2,...}), [Here we have used the where {A(kl,...,kr) } is a (suitably bounded) multiple complex sequence. parameters and ml... m r in order to identify the members of the class of the multivariable polynomials defined by (9) above.] In terms of these multivariable polynomials as the basis functions, Srivastava [11] gave three general families of polynomial expansions for a multivariable function I)(Zl"’"Zr):k1 k E k A(kl,...,kr) M z 1 1 ...z r r kr (10) 0 where M is given already with the definition (9) and {f}= 0 is a bounded sequence of essentially arbitrary complex numbers. Of our interest in the present paper is only one of these families, which we recall here in the form (cf. Srivastava [11, p. 300, Equation (1.4)]): (WmlZl"’"W zr) E rt!(/ q- n) En(;w)" (ml"’"mr;Zl"’"Zr)’ (11) n n=O where, for convenience, e=O (a + It is understood that the variables I and the polynomial expansion (11) exists. + z I,..-, Zr are so constrained that both sides of Upon substituting from (7) into the left-hand side of (11), we readily obtain (wmlZl,...,w Zr)- Sn(ml,...,mr;Zl,...,Zr)n wn, (13) MING-PO CHEN and H.M. SRIVASTAVA 426 where A(,..., )zl...z r k s,(1,..., ,%; z,..., z): mlk 1 +... + turk r (14) n (mj E N(j -1,...,r); nEN0:-NU{0}). On the other .hand, the right-hand side of (11) can easily be rewritten as wn rim0 fl.-. n (- 1 )k kin0 k) II)(ml’"" ()(A+2k)I’(A+ "’ mr; ]-)" I’(A -t- n -t- k -t- Zl"" "’ zr)" Thus, upon equating the coefficients of w n from both sides of Srivastava’s expansion (11), we find that Sn(ml,...,mr;zl,...,Zr " () n (A+2k)r(A+k) (_1) k k n!r(A+n+k+l) k=O (15) II’k)(ml, ., mr; Zl, ., Zr), where the multivariable polynomials Sn(ml,. mr; Zl, ., z r) (14). Indeed, by appealing to Dixon’s summation theorem for a well-poised Clausenian hypergeometric 3F2 series (cf., eg., Erdlyi et al. [8, p. 189, Equation 4.4(5)], it is not difficult to give a direct proof of the polynomial identity (15). are defined by In the two-variable case (r 2), if we further set m 1-1, m 2-m _ (mGN), z l-z, andz 2-, we find from (14) and (15) that [/-1 Z A(n-mk, k)zn-mk( Z (-1)k( )]n!r((A+ 2k)I’(:++ k++k)’l) II)(1 m; z, () k- k=O n k=O where [n/m] denotes, as usual, the greatest integer in n/m a two-variable polynomial given, by analogy with (9), by II’X)(1, re;z,(): (16) (n No;m e N), and II’X)(1, m; z, ) is pTmq<_k (- k)p+mq(A + k)p+ mqA(p,q)zP( q (m e N). (17) p,q--O For m- 1, the polynomials occurring on the left-hand side of (16) can be identified with the classical Jacobi polynomials if we specialize the do (...truncated)