The elliptic function ${\rm dn}_3$ of Shen

TL;DR

This paper analyzes Shen's elliptic function ${ m dn}_3$, revealing its connection to Ramanujan's elliptic functions in signature three and deriving a notable hypergeometric identity.

Contribution

It provides a detailed analysis of ${ m dn}_3$, linking it to Ramanujan's theory and deriving a significant hypergeometric identity.

Findings

Connection established between ${ m dn}_3$ and Ramanujan's elliptic functions

Derivation of a new hypergeometric identity

Enhanced understanding of elliptic functions in signature three

Abstract

We analyze the elliptic function ${\rm dn}_3$ introduced by Li-Chien Shen, contributing to the Ramanujan theory of elliptic functions in signature three. A famous hypergeometric identity emerges from our analysis.