A generalization of the cosine addition law on semigroups

Abstract

Our main result is that we describe the solutions $g,f:S\to \mathbb{C}$ of the functional equation \[g(xσ(y))=g(x)g(y)-f(x)f(y)+αf(xσ(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}$ is a fixed constant and $\sigma :S\to S$ an involutive automorphism.