- Title
- Generating functions of Legendre polynomials: a tribute to Fred Brafman
- Creator
- Wan, James; Zudilin, W.
- Relation
- Funding BodyARCGrant NumberDP110104419 http://purl.org/au-research/grants/arc/DP110104419
- Relation
- Journal of Approximation Theory Vol. 164, Issue 4, p. 488-503
- Publisher Link
- http://dx.doi.org/10.1016/j.jat.2011.12.001
- Publisher
- Academic Press
- Resource Type
- journal article
- Date
- 2012
- Description
- 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.
- Subject
- Legendre polynomial; Brafman’s generating function; hypergeometric series; Clausen's identity; Apéry like sequence
- Identifier
- http://hdl.handle.net/1959.13/934773
- Identifier
- uon:11902
- Identifier
- ISSN:0021-9045
- Language
- eng
- Full Text
- Reviewed
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