A generating function of the squares of Legendre polynomials

TL;DR

This paper connects a generating function for squared Legendre polynomials to a hypergeometric series with a modular parametrization, enabling the resolution of identities involving 1/π observed experimentally.

Contribution

It introduces a new link between Legendre polynomial generating functions and modular functions, resolving specific identities involving 1/π.

Findings

Established a relation between generating functions and hypergeometric series with modular parametrization.

Resolved a subfamily of identities involving 1/π observed by Zhi-Wei Sun.

Utilized Shaun Cooper's recent work on modular functions for this analysis.

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 Shaun Cooper. By using this modular parametrisation we resolve a subfamily of identities involving $1/\pi$ which was experimentally observed by Zhi-Wei Sun.