The theory behind solvability of some difference equations and systems

Abstract

Quite recently in the paper ‘Qualitative behavior of higher-order rational difference systems: positivity, asymptotic behavior, and periodicity’ (J. Inequal. Appl. 2025:121, 2025) have been given lists of some closed-form formulas for solutions to three concrete systems of difference equations, without any theoretical explanation how they are obtained. Here, we show that the formulas therein can be obtained by using theoretically established methods, and give some detailed explanations for how some more general difference equations and systems of difference equations can be solved in closed form. We also show that several results in the above-mentioned paper are not correct, point out some cardinal mistakes in the reasoning therein, and give some counterexamples. We also give some comments on the cited papers therein.