Solvability of the class of two-dimensional product-type systems of difference equations of delay-type (1, 3, 1, 1)

Abstract

This paper essentially presents the last and important steps in the study of (practical) solvability of two-dimensional product-type systems of difference equations of the following form zn = αzan-k wbn-l, wn = βwn-mczn-sd, n ∈ N0, where k, l, m, s ∈ N, a, b, c, d ∈ Z, and where α, β and the initial values are complex numbers. It is devoted to the most complex case which has not been considered so far (the case k = l = s = 1 and m = 3). Closed form formulas for solutions to the system are found in all possible cases. The structure of the solutions to the system is considered in detail. The following five cases: (1) b = 0; (2) c = 0; (3) d = 0; (4) ac ≠ 0; (5) a = 0, bcd ≠ 0, are considered separately. Some of the situations appear for the first time in the literature.