Modular transformations and the elliptic functions of Shen

TL;DR

This paper uses Weierstrassian modular transformations to compute fundamental periods of specific elliptic functions introduced by Shen, advancing the understanding of their modular properties.

Contribution

It introduces a method to compute fundamental periods of Shen's elliptic functions using Weierstrassian modular transformations, providing new insights into their structure.

Findings

Computed fundamental periods for ${ m dn}_2$ and ${ m dn}_3$ functions

Demonstrated the application of modular transformations to Shen's elliptic functions

Enhanced understanding of the modular properties of these functions

Abstract

We employ Weierstrassian modular transformations to compute fundamental periods for the elliptic functions ${\rm dn}_2$ and ${\rm dn}_3$ of Shen.