Bulletin of the Australian Mathematical Society

Research Article

A GENERATING FUNCTION OF THE SQUARES OF LEGENDRE POLYNOMIALS

WADIM ZUDILIN 

School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia

Abstract

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 which was experimentally observed by Sun.

(Received November 28 2012)

(Accepted February 06 2013)

(Online publication March 22 2013)

2010 Mathematics subject classification

  • primary 33C20;
  • secondary 11F03;
  • 11F11;
  • 11Y60;
  • 33C45

Keywords and phrases

  • π;
  • Legendre polynomial;
  • generating series;
  • binomial sum;
  • modular function

Footnotes

  The author was supported by the Australian Research Council.