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

다국어 초록 (Multilingual 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 $n=3$.