Vol. 29, No. 1, 1969

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A bilateral generating function for the ultraspherical polynomials

S. K. Chatterjea

Vol. 29 (1969), No. 1, 73–76
Abstract

The following differentiation formula for the Ultraspherical polynomials Pnλ(x) was given by Tricomi:

 λ ---x---    (− 1)n(x2 −-1)λ+1∕2n n 2    −λ
Pn(√x2-−-1) =         n!       D  (x − 1)  .
(1.1)

The object of this paper is to point out that the formula of Tricomi leads us to the following bilateral generating function for the Ultraspherical polynomials:

Theorem. If F(x,t) = m=0amtmPmλ(x), then

      x − t ty   ∑∞
ρ−2λF(-----,--) =   trbr(y)Pλr (x),
ρ   ρ    r=0
(1.2)

where

        ∞  (  )
br(y) = ∑   r  amym,  and ρ = (1 − 2xt+ t2)1∕2.
m=0  m

Starting from the formula (1.2), one can derive a large number of bilateral generating functions for the Ultraspherical polynomials by attributing different values to am.

Mathematical Subject Classification
Primary: 33.40
Milestones
Received: 20 April 1968
Published: 1 April 1969
Authors
S. K. Chatterjea