An extensions of Kannappan's and Van Vleck's functional equations on semigroups

TL;DR

This paper extends the analysis of Kannappan-Van Vleck functional equations on semigroups, providing solutions in terms of multiplicative functions and d'Alembert's equation, without requiring the semigroup to be abelian or unital.

Contribution

It introduces new solutions for the functional equations on general semigroups, broadening the scope beyond previous assumptions of commutativity or identity elements.

Findings

Solutions expressed via multiplicative functions and d'Alembert's equation

Applicable to non-abelian, non-unital semigroups

Provides explicit solution forms for the functional equations

Abstract

This paper treats two functional equations, the Kannppan-Van Vleck functional equation $$\mu(y)f(x\tau(y)z_0)\pm f(xyz_0) =2f(x)f(y), \;x,y\in S$$ and the following variant of it $$\mu(y)f(\tau(y)xz_0)\pm f(xyz_0) = 2f(x)f(y), \;x,y\in S,$$ in the setting of semigroups $S$ that need not be abelian or unital, $\tau$ is an involutive morphism of $S$, $\mu$ : $S\longrightarrow \mathbb{C}$ is a multiplicative function such that $\mu(x\tau(x))=1$ for all $x\in S$ and $z_0$ is a fixed element in the center of $S$. We find the complex-valued solutions of these equations in terms of multiplicative functions and solutions of d'Alembert's functional equation.