97,229 research outputs found
Generating functions of Legendre polynomials: A tribute to Fred Brafman
AbstractIn 1951, Brafman derived several âunusualâ generating functions of classical orthogonal polynomials, in particular, of Legendre polynomials Pn(x). His result was a consequence of Baileyâs identity for a special case of Appellâs hypergeometric function of the fourth type. In this paper, we present a generalisation of Baileyâs identity and its implication to generating functions of Legendre polynomials of the form ân=0âunPn(x)zn, where un is an ApĂ©ry-like sequence, that is, a sequence satisfying (n+1)2un+1=(an2+an+b)unâcn2unâ1, where nâ„0 and uâ1=0, u0=1. Using both Brafmanâs generating functions and our results, we also give generating functions for rarefied Legendre polynomials and construct a new family of identities for 1/Ï
- âŠ