Jacobi Polynomials, I. New Proofs of Koornwinder’s Laplace Type Integral Representation and Bateman’s Bilinear Sum

This is the first of a series of papers which will give simple proofs of a number of recent formulas for Jacobi polynomials. In this paper one of Gegenbauer’s proofs for his integral representation of ultraspherical polynomials is given, and then a fractional integration gives Koornwinder’s integral representation for Jacobi polynomials. This is then combined with Koornwinder’s product formula to give a new proof of a bilinear sum of Bateman.