A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [ 1, 2, 9, 10, 13-16 ]. Standard defining relation [ 7, 18 ] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function: with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials R n (α,β) ( x ), normalized to unity at x = 1, are more convenient to use: (1.1) Roughly speaking, the linearization problem consists of finding the coefficients g ( k, m, n; α,β ) in the expansion (1.2)