Cosine-sine functional equation on semigroups

Abstract

Let $S$ be a semigroup. We determine the complex-valued solutions $f,g,h$ of the functional equation \begin{equation*}f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y), x,y\in S,\end{equation*} in terms of multiplicative functions, solutions of the special case $$\phi (xy)=\phi (x)\chi (y)+\chi (x)\phi (y),x,y\in S$$ of the sine addition law, where $\chi :S\to \mathbb{C}$ is a multiplicative function, and also in terms of solutions of the particular case $$\psi (xy)=\psi (x)\chi (y)+\chi (x)\psi (y)+\phi (x)\phi (y),x,y\in S$$ of the cosine-sine functional equation where $\chi :S\to \mathbb{C}$ is a multiplicative function and $\phi :S\to \mathbb{C}$ such that the pair $(\phi ,\chi )$ satisfies the sine addition law.