A generalization of d’Alembert’s other functional equation on semigroups

A d'Alembert type functional equation on semigroups

“…The following result shows that the solutions of (1.2) on a t-compatible semigroup have the same forms found in [2,Theorem 4.3] for semigroups generated by their squares. Theorem 3.4.…”

“…on groups and monoid generated by its squares, where µ : M → C is a multiplicative function such that µ(xσ(x)) = 1 for all x ∈ M . Moreover, the solutions of (1.2) on a semigroup generated by its squares are also known (See Ajebbar and Elqorachi [2]). Recently, Ebanks [7] solved (1.1) on a larger class of monoids that contains the class of monoids generated by their squares, and regular monoids.…”

“…Stetkaer [12,Corollary 3.5] derives the solution formulas for the functional equation (1.1) on abelian groups. The method used in [2,3,7,9] proceeds from the starting point of the trigonometric functional equation µ(y)k(xσ(y)) = k(x)l(y) − k(y)l(x), x, y ∈ S.…”

A generalization of the cosine addition law on semigroups

“…The pair (f, g) on the left falls into case (2). So does the pair on the right, except that there χ is replaced by χ * .…”

“…on groups, where σ : S → S is an involutive automorphism, and τ : S → C is a multiplicative function such that τ (xσ(x)) = 1 for all x ∈ S. It was also solved on groups with τ = 1 by Fadli et al [9]. The results were extended to semigroups generated by their squares by Ajebbar and Elqorachi [2]. Recently, Aserrar, Chahbi and Elqorachi [3] obtained the solutions on semigroups.…”