A generalization of the cosine addition law on semigroups

TL;DR

This paper generalizes the cosine addition law within semigroups by characterizing solutions to a specific functional equation involving an involutive automorphism, expanding understanding of functional equations in algebraic structures.

Contribution

It provides a comprehensive description of solutions to a new functional equation on semigroups, extending classical cosine addition laws to more general algebraic settings.

Findings

Characterization of solutions to the functional equation on semigroups

Extension of cosine addition law to algebraic structures with involutive automorphisms

New insights into functional equations involving automorphisms and complex functions

Abstract

Our main result is that we describe the solutions $g,f:S\rightarrow\mathbb{C}$ of the functional equation \[g(x\sigma(y))=g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}$ is a fixed constant and $\sigma :S\rightarrow S$ an involutive automorphism.