Some trigonometric functional equations on monoids generated by their squares

The Cosine-Sine Functional Equation on Semigroups

“…The next lemma shows that the solution of (1.1) on compatible semigroups will generalize the results of [1] and [2].…”

“…The authors of [1] showed that the solutions of (1.1) on any group are exponential polynomials, whereas in [2] we showed that this is not generally the case on semigroups. For example, where a solution of (1.1) on a group contains a term Aχ, with A additive and χ exponential, on a semigroup we may see instead the term…”

“…Since the case of groups and the case of semigroups generated by their squares were handled in [1] and [2] respectively, we use semigroups which are not groups and not generated by their squares.…”

“…Equation (1.1) was solved by Chung, Kannappan, and Ng ( [1]) for the case that S is a group. Their result was extended by the author ( [2]) to the case that S is a semigroup generated by its squares. The main goal of the present work is to extend that result to a larger class of semigroups.…”

Around the Sine Addition Law and d’Alembert’s Equation on Semigroups

“…The main goal of this paper is to solve the functional equation f (xσ(y)) + h(τ (y)x) = 2f (x)k(y), x,y ∈ M, (1) for unknown functions f, h, k : M → C, where M is a monoid (i.e. a semigroup with identity element that we denote e), and σ, τ : M → M are given involutive…”