k-gamma, k-beta matrix functions and their properties

Abstract

The main aim of this paper is to define $k$-gamma and $k$-beta matrix functions, and derive the conditions for matrices $M,N$ so that the $k$-beta matrix function $B_{k}(M,N)$ satisfies the relations $B_{k}(M,N)=B_{k}(N,M)$ and $B_{k}(M,N)=\Gamma_{k}(M)\Gamma_{k}(N)\Gamma_{k}^{-1}(M+N)$ in the form of $k$-symbol, where $k>0$. A limit expression for the $k$-gamma function of a matrix is also established.