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 Pn(x)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 generalisation of Bailey’s identity and its implication to generating functions of Legendre polynomials of the form ∑n=0∞unPn(x)zn, where unun is an Apéry-like sequence, that is, a sequence satisfying (n+1)2un+1=(an2+an+b)un−cn2un−1(n+1)2un+1=(an2+an+b)un−cn2un−1, where n≥0n≥0 and u−1=0u−1=0, u0=1u0=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/π1/π.