A class of functional equations on monoids

TL;DR

This paper extends the analysis of a class of functional equations on monoids and groups, providing solutions for more general equations involving involutive automorphisms and multiplicative functions, with applications to related equations.

Contribution

It generalizes previous results by solving a broader class of functional equations involving multiplicative functions and involutive automorphisms on monoids and groups.

Findings

Derived explicit solutions for the generalized functional equations.

Extended the solution framework to include multiplicative functions with specific properties.

Applied the results to solve related equations on groups.

Abstract

In \cite{05} B. Ebanks and H. Stetk{\ae}r obtained the solutions of the functional equation $f(xy)-f(\sigma(y)x)=g(x)h(y)$ where $\sigma$ is an involutive automorphism and $f,g,h$ are complex-valued functions, in the setting of a group $G$ and a monoid $M$. Our main goal is to determine the complex-valued solutions of the following more general version of this equation, viz $f(xy)-\mu(y)f(\sigma(y)x)=g(x)h(y)$ where $\mu: G\longrightarrow \mathbb{C}$ is a multiplicative function such that $\mu(x\sigma(x))=1$ for all $x\in G$. As an application we find the complex-valued solutions $(f,g,h)$ on groups of the equation $f(xy)+\mu(y)g(\sigma(y)x)=h(x)h(y)$.