A fully pexiderized variant of d'Alembert's functional equations on monoids
The functional equation \[f(xy)+g(\sigma(y)x)=h(x)k(y)\] in which \(f, g, h, k\) are complex-valued functions on groups or monoids generated by their squares and \(\sigma\) is an involutive automorphism is a pexiderization of both d'Alembert functional equations \(g(x+y)+g(x-y)=2g(x)g(y)\) and \(f(x+y)-f(x-y)=g(x)h(y)\) and a a variant of the Wilson functional equation \(f(x+y)+g(x-y)=h(x)k(y)\) for monoids. The author solves the above pexiderized functional equation. The special case of the equation when \(h = 2f\) was recently investigated by him in [Aequationes Math. 95, No. 2, 383--391 (2021; Zbl 1462.39021)]. Furthermore, he explores the continuous solutions on topological groups and monoids.
scientific article; zbMATH DE number 7362283