A variant of d'Alembert functional equation on monoids
In this paper, we determine the complex-valued solutions of the functional equation $$ f(xσ(y))+f(τ(y)x)=2f(x)f(y)$$ for all $x,y \in M$, where $M$ is a monoid, $σ$: $M\longrightarrow M$ is an involutive automorphism and $τ$: $M\longrightarrow M$ is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of $2$-dimensional irreducible representations of $M$.
math.GM↗