A class of functional equations on monoids

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 $\mu :G⟶\mathbb{C}$ 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 $f(xy)+\mu (y)g(\sigma (y)x)=h(x)h(y)$.