A generalization of the symmetrized multiplicative Cauchy equation

• DOI: 10.1007/s10474-016-0584-3
• Publication Date: 2016-02-20T10:22:24Z
• Authors: A. Chahbi; B. Fadli; S. Kabbaj
• Keywords: 0101 mathematics; 01 natural sciences

Let S be a semigroup, and let \({\sigma,\tau \in {\rm Hom}(S,S)}\) satisfy \({\tau\circ\tau = \sigma\circ\sigma = \rm{id}}\). We determine the solutions \({f : S \to \mathbb{C}}\) of the functional equation $$f(x\sigma(y)) + f(\tau(y)x) = 2f(x)f(y),\quad x,y \in S,$$ in terms of multiplicative functions on S.