Generating functions of Legendre polynomials: a tribute to Fred Brafman

In 1951, Brafman derived several “unusual” generating functions of classical orthogonal polynomials, in particular, of Legendre polynomials P_n(x). His result was a consequence of Bailey’s identity for a special case of Appell’s hypergeometric function of the fourth type. In this paper, we present a generalisation of Bailey’s identity and its implication to generating functions of Legendre polynomials of the form Σ^∞/_n=0 u_nP_n(x)zⁿ, where u_n is an Apéry-like sequence, that is, a sequence satisfying (n + 1)²u_n+1 = (an² + an + b)u_n − cn²u_n−1, where n ≥ 0 and u₋₁ = 0, u₀ = 1. Using both Brafman’s generating functions and our results, we also give generating functions for rarefied Legendre polynomials and construct a new family of identities for 1/π.