Generating functions of Legendre polynomials: A tribute to Fred Brafman. (English) Zbl 1242.33018

Wan, James; Zudilin, Wadim

J. Approx. Theory 164, No. 4, 488-503 (2012).

The authors present a generalization of Bailey’s identity for Appell’s hypergeometric function and its implication to generating functions of Legendre polynomials of the form \(\sum_{n=0}^{\infty}u_{n}P_{n}(x)z^{n}\), where \(u_{n}\) is an Apéry-like sequence, that is, a sequence satisfying \((n+1)^2u_{n+1}=(an^2+an+b)u_n-cn^2u_{n-1}\), where \(n\geq 0\) and \(u_{-1}=0, u_0=1\). The authors also give generating functions for rarefied Legendre polynomials and construct a new family of identities for \(1/\pi\).

Reviewer: Stamatis Koumandos (Nicosia)

MSC:

33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C05	Classical hypergeometric functions, \({}_2F_1\)
33C20	Generalized hypergeometric series, \({}_pF_q\)

Keywords:

Legendre polynomial; Brafman’s generating function; hypergeometric series; Clausen’s identity; Apéry-like sequence; modular function; series for \(\pi \)

References:

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