The Cosine–Sine functional equation on a semigroup with an involutive automorphism

Abstract

Let S be a semigroup. We determine the complex-valued solutions f, g, h of the functional equationin terms of multiplicative functions, solutions of the special caseof the sine addition law, where χ : S → C is a multiplicative function, and also in terms of solutions of the particular caseof the cosine-sine functional equation where χ : S → C is a multiplicative function and ϕ : S → C such that the pair (ϕ, χ) satisfies the sine addition law.

A generalization of the cosine addition law on semigroups

“…In [12] Vincze obtained the solutions of (1.2) on abelian groups with σ = id, and it was solved on general groups by Chung, Kannappan, and Ng [4]. The results were extended to the case of topological groups by Poulsen and Stetkaer [8] and to semigroups generated by their squares by Ajebbar and Elqorachi [2] with σ an involutive automorphism. Also Stetkaer [11,Theorem 6.1] gives a description of the solution of (1.1) with σ = id on a general semigroup in terms of the solutions of (1.3) with σ = id.…”

Stability of the Cosine-Sine Functional Equation on amenable groups

“…Chung et al [4] solved the functional equation (1.1) on groups. Recently, Ajebbar and Elqorachi [2] obtained the solutions of the functional equation (1.1) on a semigroup generated by its squares. The stability properties of the functional equations (1.2) and (1.3) have been obtained by Székelyhidi [13] on amenable groups.…”

Extensions of the Cosine-Sine functional equation

“…Chung, Kannappan and Ng [5] solved the functional equation f (xy) = f (x)g(y) + f (y)g(x) + h(x)h(y), x, y ∈ G, where G is a group. Recently, the results of [5] were extended by the authors [2] to semigroups generated by their squares.…”