A variant of Wilson's functional equation on semigroups

TL;DR

This paper characterizes solutions to a variant of Wilson's functional equation on semigroups involving automorphisms and multiplicative functions, expanding understanding of functional equations in algebraic structures.

Contribution

It provides a complete description of complex solutions to a new form of Wilson's functional equation on semigroups with automorphisms.

Findings

Explicit solutions characterized for the functional equation

Conditions on multiplicative functions and automorphisms identified

Generalization of Wilson's equation to semigroup context

Abstract

We determine the complex-valued solutions of the following functional equation \[f(xy)+\mu (y)f(\sigma (y)x) = 2f(x)g(y),\quad x,y\in S,\] where $S$ is a semigroup and $\sigma$ an automorphism, $\mu :S\rightarrow \mathbb{C}$ is a multiplicative function such that $\mu (x\sigma (x))=1$ for all $x\in S$.