Local Identities Involving Jacobi Elliptic Functions

TL;DR

This paper develops new local identities involving Jacobi elliptic functions, simplifies existing cyclic identities, generalizes them with phase factors, and connects these identities to discretizations of nonlinear differential equations, providing explicit integral solutions.

Contribution

It introduces a systematic approach to local identities of Jacobi elliptic functions, extending cyclic identities and linking them to differential equations and integral evaluations.

Findings

Derived new local identities of arbitrary rank.

Extended cyclic identities with phase factors.

Connected identities to discretizations of nonlinear equations.

Abstract

We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2i\pi/s), where s is any integer. Third, we systematize the local identities by deriving four local ``master identities'' analogous to the master identities for the cyclic sums discussed by us previously. Fourth, we point out that many of the local identities can be thought of as exact discretizations of standard nonlinear differential equations satisfied by the Jacobian elliptic functions. Finally, we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving Jacobi elliptic functions.