Vol. 35, No. 3, 1970

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ISSN: 0030-8730
A general theorem for bilinear generating functions

S. Saran

Vol. 35 (1970), No. 3, 783–786
Abstract

The following theorem was proved by Chatterjea for ultraspherical polynomials:

            ∞∑     m  λ
If F (x,t) =    amt  Pm(x)
m=0

then

-2λ  x − t ty   ∞∑  r      λ
ρ  F(--ρ-,-ρ ) =   tbr(y)P r (x)
r=0

where br(y) = m=0(vm)amym and ρ = (1 2xt + t2)12. The object of this paper is to show that a general theorem for any polynomial satisfying certain conditions can be given so as to include the above case, and may be applicable in obtaining new generating functions for other polynomials also.

Theorem. If

f (x) = μ(n)G(x)Dn {g(x)}
n
(1.1)

where g(x) and G(x) are independent of n, and

        ∑∞
F(x,t) =    amtmfm (x )
m=0
(1.2)

then

                 ∞
G-(x)F-(x-−-t,ty)  ∑   (− t)r
G (x− t)   =  r=0 μ(r)r!br(y)fr(x)

where

       ∑r              m
br(y) =   (− r)m μ(m)amy
m=0

Mathematical Subject Classification
Primary: 33.20
Milestones
Received: 7 April 1970
Published: 1 December 1970
Authors
S. Saran