New hypergeometric formulae to $\pi$ arising from M. Roberts hyperelliptic reductions

TL;DR

This paper develops new hypergeometric identities for π using Roberts's hyperelliptic reductions, expressing π through elliptic, hypergeometric, and Gamma functions with novel formulas.

Contribution

It introduces new identities for π derived from hyperelliptic reductions, expanding the mathematical tools for computing special functions.

Findings

New formulas for π involving elliptic, hypergeometric, and Gamma functions.

Expressions of π with one or two parameters and the imaginary unit.

Potential applications in verifying computational routines for Lauricella's functions.

Abstract

In this article we developed a special topic of our pure-mathematics papers concerning the hypergeometric theory. Based upon a Roberts's reduction approach of hyperelliptic integrals to elliptic ones and on the simultaneous multivariable hypergeometric series evaluation of them, several identities have been obtained expressing $\pi$ in terms of special values of elliptic, hypergeometric and Gamma functions. By them $\pi$ can be provided through either only one or two parameters and through the imaginary unit. In any case, such results, all unpublished and undoubtably new, will provide, beyond their own beauty, a useful tool in order to check the routines (more or less naive) which one can build for the practical computations of Lauricella's functions met frequently in researches on Mechanics or Elasticity.