Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(k−m)+f(k) with commutative matrices

In the investigation performed we give, on half-infinity discrete intervals, formulas for solution of initial problem of linear discrete systems x(k+1)=Ax(k)+Bx(k−m)+f(k) with constant square matrices A, B such that AB=BA, detA≠0 and with a vector function f(k). Corresponding representations are obtained with the aid of so-called discrete matrix delayed exponential, which permits to represent solutions in a matrix form similarly as for ordinary differential systems with constant matrices, or as well as for differential systems with constant matrices and constant delay.