Around the sine addition law and d'Alembert's equation on semigroups (Q2051430)

The author solves the partially pexiderized d'Alembert-type functional equation \(f(x\sigma(y))+h(\tau(y)x)=2f(x)k(y)\), where \(f, h, k\) are complex-valued functions on a monoid \(M\), i.e., a semigroup with identity element, and \(\sigma\) and \(\tau\) are given involutive automorphisms. The genealogy of this equation traces back to the d'Alembert functional equation \(f(x+y)+f(x-y)=2f(x)f(y)\), the Wilson functional equation \(f(x+y)+g(x-y)=h(x)k(y)\), as well as the symmetrized multiplicative Cauchy equation \(f (xy) + f (yx) = 2f (x)f (y)\). Among other things, he gives the continuous solution of the above functional equations on topological semigroups and monoids.