Elliptic functions from hypergeometric integrals

TL;DR

This paper investigates the connection between hypergeometric integrals and elliptic functions, identifying specific signatures where Jacobian analogues are truly elliptic, advancing the understanding of elliptic functions in alternative bases.

Contribution

It precisely determines the signatures in which Jacobian analogues derived from hypergeometric integrals are genuinely elliptic functions.

Findings

Identifies signatures where Jacobian analogues are elliptic

Extends Ramanujan's theory of elliptic functions to new bases

Clarifies the conditions for ellipticity of hypergeometric-derived functions

Abstract

As a contribution to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has shown how analogues of the Jacobian elliptic functions may be derived from incomplete hypergeometric integrals in signatures three and four. We determine precisely the signatures in which the Jacobian analogues or their squares are indeed elliptic.