A variant of d’Alembert’s functional equation

Abstract

Let S be a semigroup, and let ${\sigma \in {\rm Hom}(S,S)}$ satisfy ${\sigma \circ \sigma = {\rm id}}$ . We show that any solution ${g: S \to \mathbb{C}}$ of the functional equation

$$ g(xy) + g(\sigma(y)x) = 2g(x)g(y), \quad x, y \in S, $$

has the form ${g = (\mu + \mu \circ \sigma) /2}$ , where μ is a multiplicative function on S. From this we find the solutions ${f: I \times I \to \mathbb{C}}$ , where I is a semigroup, of

$$ f(pr, qs) + f(sp, rq) = f(p, q)f(r, s), \quad p, q, r, s \in I, $$

thereby generalizing a result by Chung, Kannappan, Ng and Sahoo for the multiplicative semigroup I = ]0, 1[.