This paper, pursuing the work started in two previous papers appeared on this journal, holds six new formulae for pi 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.

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

MINGARI SCARPELLO, GIOVANNI;RITELLI, DANIELE
2014

Abstract

This paper, pursuing the work started in two previous papers appeared on this journal, holds six new formulae for pi 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.
Giovanni Mingari Scarpello; Daniele Ritelli
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/237482
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