Representation of the solutions of linear discrete systems with constant coefficients and two delays (Q1723863)

Summary: The purpose of this paper is to develop a method for the construction of solutions to initial problems of linear discrete systems with constant coefficients and with two delays \(\Delta x(k)=Bx(k-m)+Cx(k-n)+f(k)\), where \(m,n\in\mathbb N\), \(m\neq n\), are fixed, \(k=0,\dots,\infty\), \(B=(b_{ij})\), \(C=(c_{ij})\) are constant \(r\times r\) matrices, \(f\) is a given \(r\times1\) vector, and \(x\) is an \(r\times1\) unknown vector. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential for two delays. Such approach results in a possibility to express an initial Cauchy problem in a closed form. Examples are shown illustrating the results obtained.