Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2024; 39(3): 785-802

Online first article July 11, 2024      Printed July 31, 2024

https://doi.org/10.4134/CKMS.c220073

Copyright © The Korean Mathematical Society.

A variant of d'Alembert's and Wilson's functional equations for matrix valued functions

Abdellatif Chahbi, Mohamed Chakiri, Elhoucien ELqorachi

Ibn Zohr University; Ibn Zohr University; Ibn Zohr University

Abstract

Given $M$ a monoid with a neutral element $e$. We show that the solutions of d'Alembert's functional equation for $n\times n$ matrices \begin{equation*} \Phi(pr,qs)+\Phi(sp,rq)=2\Phi(r,s)\Phi(p,q),\quad p,q,r,s\in M \end{equation*} are abelian. Furthermore, we prove under additional assumption that the solutions of the n-dimensional mixed vector-matrix Wilson's functional equation \begin{equation*} \left\lbrace\begin{array}{ll} f(pr,qs)+f(sp,rq)=2\Phi(r,s)f(p,q),\\ \Phi(p,q)=\Phi(q,p),\quad p,q,r,s\in M \end{array}\right. \end{equation*} are abelian. As an application we solve the first functional equation on groups for the particular case of $n=3$.

Keywords: Topological group, monoid, D'Alembert's equation, Wilson's equation, matrix, automorphism, involution

MSC numbers: 39B52, 39B42, 39B32