A Generalization of Landen's Quadratic Transformation Formulas for Jacobi Elliptic Functions

TL;DR

This paper generalizes Landen's quadratic transformation formulas for Jacobi elliptic functions, using recent solutions of nonlinear differential equations and new cyclic identities, expanding their applicability.

Contribution

It introduces significant generalizations of Landen's formulas based on novel cyclic identities and solutions of nonlinear differential equations.

Findings

New cyclic identities involving Jacobi elliptic functions.

Generalized Landen formulas for broader parameter ranges.

Connections to solutions of nonlinear differential equations.

Abstract

Landen formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by making a suitable quadratic transformation of variables in elliptic integrals. We obtain and discuss significant generalizations of the celebrated Landen formulas. Our approach is based on some recently obtained periodic solutions of physically interesting nonlinear differential equations and numerous remarkable new cyclic identities involving Jacobi elliptic functions.