A system of cosine-sine functional equations on a semigroup generated by its squares

TL;DR

This paper characterizes the solutions of a system of cosine-sine functional equations on a semigroup generated by its squares, expanding understanding of such equations in algebraic structures.

Contribution

It provides a complete solution classification for the system of cosine-sine functional equations on semigroups generated by squares, a novel algebraic setting.

Findings

Solutions are explicitly characterized for the given system.

The structure of solutions depends on the constants , .

The results extend previous work on functional equations in semigroups.

Abstract

Given a semigroup $S$ generated by its squares, we determine the complex-valued solutions of the following system of cosine-sine functional equations \begin{align*} f(xy)=f(x)g_{1}(y)+g_{1}(x)f(y)+\lambda_{1}^{2}\,h(x)h(y),\; x,y\in S,\\ h(xy)=h(x)g_{2}(y)+g_{2}(x)h(y)+\lambda_{2}^{2}\,f(x)f(y),\; x,y\in S, \end{align*} where $\lambda_{1},\lambda_{2}\in\mathbb{C}$ are given constants and $f,g_{1},g_{2},h:S\to\mathbb{C}$ are unknown functions.