A generalization of the cosine addition law on semigroups

Abstract

Let S be a semigroup. Our main results is that we describe the complex-valued solutions of the following functional equationswhere σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications.

The equation f(xy) = f(x)h(y) + g(x)f(y) and representations on C2

“…The sine subtraction law which is named after the formula sin(x − y) = sin x cos y − cos x sin y where x, y ∈ R, from elementary trigonometry, has a long history, but new facets of it are still being discovered. Thus Aserrar and Elqorachi [3,4] and Ebanks [8,9,11] have recently studied various extensions of it from R to semigroups. For the classic case of an abelian group see Aczél and Dhombres [1, pp.…”