Cyclic Identities Involving Jacobi Elliptic Functions. II

TL;DR

This paper derives fundamental cyclic identities involving Jacobi elliptic functions, providing a basis for solving nonlinear equations and extending to complex shifts and theta functions.

Contribution

The authors introduce four master identities that generalize previous cyclic sum identities for Jacobi elliptic functions, including cases with alternating signs and complex shifts.

Findings

Derived four master identities for cyclic sums of Jacobi elliptic functions.

Extended identities to include alternating signs and complex shifts.

Connected identities to ratios of Jacobi theta functions.

Abstract

Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at $p$ equally shifted points on the real axis were recently found. These identities played a crucial role in discovering linear superposition solutions of a large number of important nonlinear equations. We derive four master identities, from which the identities discussed earlier are derivable as special cases. Master identities are also obtained which lead to cyclic identities with alternating signs. We discuss an extension of our results to pure imaginary and complex shifts as well as to the ratio of Jacobi theta functions.