The solution of a generalized variant of d’Alembert’s functional equation

Abstract

Let $(S,\cdot )$ be a semigroup, $\mathbb {C}$ be the set of complex numbers, and let $\sigma ,\tau \in Hom(S,S)$ satisfy $\tau \circ \tau =\sigma \circ \sigma =id.$ We show that any solution $f:S \rightarrow \mathbb {C}$ of the functional equation

$$\begin{aligned} f(x\sigma (y))+\chi (y)f(\tau (y)x)=2f(x)f(y), \quad x,y \in S, \end{aligned}$$

has the form $f=(m+\chi \, m\circ \sigma \circ \tau )/2$, where m is a multiplicative function on S and $\chi :S\rightarrow (\mathbb {C}\backslash \{0\},\cdot )$ is a character on S (i.e., $\chi (xy)=\chi (x)\chi (y)$ for all $x,y\in S$) which satisfies $\chi (x\tau (x))=1$ for all $x\in S$.