Cyclic Identities Involving Jacobi Elliptic Functions

TL;DR

This paper presents numerous cyclic identities involving Jacobi elliptic functions with arguments separated by fractions of the complete elliptic integral, providing algebraic and numerical methods for their verification and exploring their polynomial structures.

Contribution

It introduces a systematic study of cyclic identities involving Jacobi elliptic functions with arguments spaced by fractions of 2K(m)/p or 4K(m)/p, including methods for their algebraic and numerical verification.

Findings

Identifies cyclic identities for various p and r values.

Provides algebraic derivations for small p and r.

Uses software to verify higher-order identities.

Abstract

We state and discuss numerous mathematical identities involving Jacobi elliptic functions sn(x,m), cn(x,m), dn(x,m), where m is the elliptic modulus parameter. In all identities, the arguments of the Jacobi functions are separated by either 2K(m)/p or 4K(m)/p, where p is an integer and K(m) is the complete elliptic integral of the first kind. Each p-point identity of rank r involves a cyclic homogeneous polynomial of degree r (in Jacobi elliptic functions with p equally spaced arguments) related to other cyclic homogeneous polynomials of degree r-2 or smaller. Identities corresponding to small values of p,r are readily established algebraically using standard properties of Jacobi elliptic functions, whereas identities with higher values of p,r are easily verified numerically using advanced mathematical software packages.