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) .](a070x.png) | (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=0∞amtmPmλ(x), then
![x − t ty ∑∞
ρ−2λF(-----,--) = trbr(y)Pλr (x),
ρ ρ r=0](a071x.png) | (1.2) |
where
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.
|