The General Analytic Solution of a Functional Equation of Addition Type

Abstract

The general analytic solution to the functional equation $$ ϕ_1(x+y)= { { \biggl|\matrix{ϕ_2(x)&ϕ_2(y)\crϕ_3(x)&ϕ_3(y)\cr}\biggr|} \over { \biggl|\matrix{ϕ_4(x)&ϕ_4(y)\crϕ_5(x)&ϕ_5(y)\cr}\biggr|} } $$ 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 $$ ϕ_1(x+y)=ϕ_4(x)ϕ_5(y)+ϕ_4(y)ϕ_5(x) $$ and \[ Ψ_1(x+y)=Ψ_2(x+y) ϕ_2(x)ϕ_3(y) +Ψ_3(x+y) ϕ_4(x)ϕ_5(y). \]