Representation of solutions to a linear matrix discrete equation with single delay

A linear matrix delayed discrete equation $\Delta X\left(k\right)=BX(k-m)+X(k-m)C+F\left(k\right),$is considered, where $m$ is a fixed positive integer, $k=0,1,\dots$ is a discrete independent variable, $X\left(k\right)$ is an $n\times n$ unknown variable matrix, $\Delta$ is the first order forward difference, $B$, $C$ are given $n\times n$ constant matrices, and $F\left(k\right)$ is a given $n\times n$ variable matrix. Using a special matrix function, under some commutativity conditions, formulas are derived, solving an initial problem $X\left(k\right)=\Psi \left(k\right)$, $k=-m,-m+1,\dots ,0$, where $\Psi \left(k\right)$ is an $n\times n$ variable matrix. Some connections with previously known results are discussed with open problems suggested for further investigations.