A variant of d’Alembert’s functional equation (original) (raw)

2014, Aequationes mathematicae

Let S be a semigroup, and let σ ∈ Hom(S, S) satisfy σ • σ = id. We show that any solution g : S → C of the functional equation g(xy) + g(σ(y)x) = 2g(x)g(y), x, y ∈ S, has the form g = (µ + µ • σ)/2, where µ is a multiplicative function on S. From this we find the solutions f : I × I → C, where I is a semigroup, of f (pr, qs) + f (sp, rq) = f (p, q)f (r, s), p, q, r, s ∈ I, thereby generalizing a result by Chung, Kannappan, Ng and Sahoo for the multiplicative semigroup I = ]0, 1[.