Generating functions of Legendre polynomials: A tribute to Fred Brafman

In 1951, F. Brafman derived several “unusual” generating functions of classical orthogonal polynomials, in particular, of Legendre polynomials Pn(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 generalization of Bailey’s identity and its implication to generating functions of Legendre polynomials of the form ∑∞ n=0 unPn(x)z , where un is an Apéry-like sequence, that is, a sequence satisfying (n + 1)un+1 = (an 2 + an + b)un − cnun−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.