Legendre Hyperelliptic integrals, {\pi} new formulae and Lauricella functions through the elliptic singular moduli

TL;DR

This paper derives six new formulas for pi using ratios of elliptic integrals and Lauricella hypergeometric functions, expanding the understanding of hyperelliptic integrals through Legendre's methods.

Contribution

It introduces six novel pi formulas by reducing hyperelliptic integrals to elliptic forms and evaluating hypergeometric Lauricella functions.

Findings

Six new formulas for pi derived from hyperelliptic integrals.

Evaluation of hypergeometric Lauricella functions at specific points.

Reduction of hyperelliptic integrals to elliptic integrals using Legendre's methods.

Abstract

This paper, pursuing the work started in [10] and [11], holds six new formulae for {\pi}, see equations, through ratios of first kind elliptic integrals and some values of hypergeometric functions of three or four variables of Lauricella type. This will be accomplished by reducing some hyperelliptic integrals to elliptic by the methods taught by Legendre in his treatise. Eventually, evaluating some hyperelliptic integrals by means of hypergeometric Lauricella functions, we obtain some further evaluations of themselves in some particular points and also in their analytic continuation.