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A GENERATING FUNCTION OF THE SQUARES OF LEGENDRE POLYNOMIALS

Part of: Discontinuous groups and automorphic forms Computational number theory

Published online by Cambridge University Press:  22 March 2013

WADIM ZUDILIN
WADIM ZUDILIN*
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia
*
[email protected]
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Abstract

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We relate a one-parametric generating function for the squares of Legendre polynomials to an arithmetic hypergeometric series whose parametrisation by a level 7 modular function was recently given by Cooper. By using this modular parametrisation we resolve a subfamily of identities involving $1/ \pi $ which was experimentally observed by Sun.

Keywords

πLegendre polynomialgenerating seriesbinomial summodular function

MSC classification

Secondary: 33C20: Generalized hypergeometric series, $_pF_q$ 11F03: Modular and automorphic functions 11F11: Holomorphic modular forms of integral weight 11Y60: Evaluation of constants 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 125 - 131
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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