The solution of the cosine-sine functional equation on semigroups (Q6168135)

Consider a topological semigroup \(S\) (the topology can be discrete) and three continuous mappings \(f,g,h\in S\to\mathbb{C}\). The following \textit{cosine-sine functional equation} is studied: \[ f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y),\qquad x,y\in S. \] For \(S\) being a {group}, the above equation was solved by \textit{J. K. Chung} et al. [Linear Algebra Appl. 66, 259--277 (1985; Zbl 0564.39002)]. Later on, solutions were given by various authors for special classes of semigroups. In the present paper, the general continuous solutions for an \textit{arbitrary} semigroup \(S\) are given. As opposed to the group case, the description of the class of solutions is far more complex and involves not only additive and multiplicative functions, but also already known solutions of other related equations.