The General Analytic Solution of a Functional Equation of Addition Type

TL;DR

This paper provides a comprehensive analytic solution to a specific addition-type functional equation, characterizing solutions using Weierstrass elliptic functions and illustrating with classical elliptic function theorems.

Contribution

It offers the first complete analytic characterization of solutions to a complex addition-type functional equation using elliptic functions.

Findings

Solutions are described in terms of Weierstrass elliptic functions.

The theory applies to classical addition theorems of Jacobi elliptic functions.

It extends to other functional equations involving elliptic functions.

Abstract

The general analytic solution to the functional equation $$ \phi_1(x+y)= { { \biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over { \biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } $$ is characterised. Up to the action of the symmetry group, this is described in terms of Weierstrass elliptic functions. We illustrate our theory by applying it to the classical addition theorems of the Jacobi elliptic functions and the functional equations $$ \phi_1(x+y)=\phi_4(x)\phi_5(y)+\phi_4(y)\phi_5(x) $$ and \[ \Psi _1(x+y)=\Psi _2(x+y) \phi_2(x)\phi_3(y) +\Psi_3(x+y) \phi_4(x)\phi_5(y). \]