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 Rn
(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)