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

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.