The cosine-sine functional equation on a semigroup with an involutive automorphism
The authors investigate the functional equation \[ f(x\sigma(y))=f(x)g(y)+g(x)f(y)+h(x)h(y), \quad x,y \in S,\tag{E} \] where \(S\) is a semigroup generated by its quares and \(\sigma\) is an involutive automorphism of \(S\). This functional equation is a generalization of sine and cosine addition laws and is a special Levi-Civta equation. The main result of the paper gives the form of the solutions of equation (E), where \(f\), \(g\) and \(h\) are complex-valued functions, in term of multiplicative and additive functions. Different treatment is required whether the functions \(f\) and \(h\) are lineraly dependent or independent.
scientific article; zbMATH DE number 6825377