Legendre hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli

Abstract

This paper, pursuing the work started by tha authots, holds six new formulae for π through ratios of first kind complete 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 through the methods Legendre taught in his treatise. The complete elliptic integrals of first kind have complementary moduli: as a consequence we can find their ratio through the Lauricella functions. In such a way we succeed in obtaining, through the theory of elliptic singular moduli, some particular values of Lauricella's themselves.