Five Trigonometric addition laws on semigroups

TL;DR

This paper solves complex-valued functional equations involving semigroup operations, involutive automorphisms, and trigonometric-like addition laws, expanding understanding of such equations in algebraic structures.

Contribution

It provides a comprehensive characterization of solutions to five new functional equations involving semigroups and involutive automorphisms, introducing novel addition laws.

Findings

Explicit solutions for each functional equation are derived.

The solutions generalize classical trigonometric addition formulas.

The work extends the theory of functional equations on algebraic structures.

Abstract

In this paper, we determine the complex-valued solutions of the following functional equations \[g(x\sigma (y)) = g(x)g(y)+f(x)f(y),\quad x,y\in S,\]\[f(x\sigma (y)) = f(x)g(y)+f(y)g(x),\quad x,y\in S,\]\[f(x\sigma (y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\quad x,y\in S,\]\[f(x\sigma(y))=f(x)g(y)+f(y)g(x)+\alpha g(x\sigma(y)),\quad x,y\in S,\]\[f(x\sigma(y))=f(x)g(y)-f(y)g(x)+\alpha g(x\sigma(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}\backslash \lbrace 0\rbrace$ is a fixed constant and $\sigma :S\rightarrow S$ an involutive automorphism.