A generalization of the cosine-sine functional equation on a semigroup

Abstract

Given a semigroup S equipped with an involutive automorphism \(\sigma \), we determine the complex-valued solutions f, g, h of the functional equation

$$\begin{aligned}f(x\sigma (y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\,\,x,y\in S,\end{aligned}$$

in terms of multiplicative functions and solutions of the special cases of sine and cosine–sine functional equations

$$\begin{aligned} \varphi (xy)=\varphi (x)\chi (y)+\chi (x)\varphi (y), x,y\in S \end{aligned}$$

and

$$\begin{aligned} \psi (xy)=\psi (x)\chi (y)+\chi (x)\psi (y)+\varphi (x)\varphi (y), x,y\in S \end{aligned}$$

where \(\chi :S\rightarrow \mathbb {C}\) is a multiplicative function.