An extension of Van Vleck's functional equation for the sine

TL;DR

This paper extends Van Vleck's functional equation for the sine by incorporating multiplicative functions and involutive automorphisms, providing solutions for these generalized equations on semigroups and monoids.

Contribution

It introduces and solves a generalized form of Van Vleck's functional equation involving multiplicative functions and involutive automorphisms.

Findings

Derived solutions for the extended functional equation on semigroups.

Solved a variant of the equation on monoids with involutive automorphisms.

Established conditions for the solutions involving multiplicative functions.

Abstract

In \cite{St3} H. Stetk\ae r obtained the solutions of Van Vleck's functional equation for the sine $$f(x\tau(y)z_0)-f(xyz_0) =2f(x)f(y),\; x,y\in G,$$ where $G$ is a semigroup, $\tau$ is an involution of $G$ and $z_0$ is a fixed element in the center of $G$. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck's functional equation for the sine $$\mu(y)f(x\tau(y)z_0)-f(xyz_0) =2f(x)f(y), \;x,y\in G,$$ where $\mu$ : $G\longrightarrow \mathbb{C}$ is a multiplicative function such that $\mu(x\tau(x))=1$ for all $x\in G$. Furthermore, we obtain the solutions of a variant of Van Vleck's functional equation for the sine $$\mu(y)f(\sigma(y)xz_0)-f(xyz_0) = 2f(x)f(y), \;x,y\in G$$ on monoids, and where $\sigma$ is an automorphism involutive of $G$.