In 1951, 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 generalisation of Bailey’s identity and its implication to generating functions of Legendre polynomials of the form Σ∞/n=0 unPn(x)zn, where un is an Apéry-like sequence, that is, a sequence satisfying (n + 1)2un+1 = (an2 + an + b)un − cn2un−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.
Journal of Approximation Theory Vol. 164, Issue 4, p. 488-503