Cyclic Identities Involving Ratios of Jacobi Theta Functions

TL;DR

This paper re-expresses cyclic identities involving Jacobi elliptic functions as ratios of Jacobi theta functions to facilitate their use in physics, providing a more accessible formulation.

Contribution

It introduces a new representation of cyclic identities using Jacobi theta functions, bridging a gap between elliptic functions and theta function applications.

Findings

Cyclic identities are reformulated in terms of Jacobi theta functions.

The new expressions are more suitable for physicists' applications.

The paper enhances the usability of elliptic function identities in physics.

Abstract

Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at $p$ equally shifted points were recently found. The purpose of this paper is to re-express these cyclic identities in terms of ratios of Jacobi theta functions, since many physicists prefer using Jacobi theta functions rather than Jacobi elliptic functions.