Hypergeometric identities in elliptic signature six

TL;DR

This paper explores hypergeometric identities related to elliptic functions in signature six, building on Ramanujan's theories and Shen's constructions, revealing new non-elliptic functions that produce these identities.

Contribution

It extends Ramanujan's elliptic function theories to signature six, identifying non-elliptic functions that generate hypergeometric identities.

Findings

Construction of non-elliptic functions in signature six

Derivation of hypergeometric identities from these functions

Connection to Ramanujan's elliptic function theories

Abstract

Within the Ramanujan theories of elliptic functions, Li-Chien Shen constructed natural elliptic functions in signature three and signature four. When applied in signature six, the same constructions produce non-elliptic functions that nevertheless engender the corresponding hypergeometric identities of Ramanujan.