Jacobi Polynomials, II. An Analytic Proof of the Product Formula

An analytic proof is given for the author’s product formula for Jacobi polynomials and a new integral representation is obtained for the product J_α (x)J_β (y) of two Bessel functions. Similarly, a product formula for Jacobi polynomials due to Dijksma and the author is derived in an analytic way. The proofs are based on Bateman’s work on special solutions of the biaxially symmetric potential equation. The paper concludes with new proofs for Gasper’s evaluation of the convolution kernel for Jacobi series and for Watson’s evaluation of the integral\[∫_0^∞ {J_α (λ x)J_β (λ y)J_β (λ z)λ 1 - α dλ .} \]