Uniqueness criterion and Cramer's rule for implicit higher order linear difference equations over \(\mathbf{Z} \) (Q2050485)

The authors study a higher-order linear difference equation, which is implicit (the coefficient of the highest indexed term is not \(\pm1\)), over the ring of integers, and over the ring of \(p\)-adic integers. In both cases they determine conditions under which the equation has a unique solution over the ring: an explicit constant sequence in the former case, and a convolution of the non-homogeneous term with a solution of a dual equation in the latter. The authors also determine conditions under which the solution can be found using an analogue of Cramer's rule. The results of this paper generalise those of \textit{V. N. Berestovskij} and \textit{Yu. G. Nikonorov} [Mat. Tr. 10, No. 1, 97--131 (2007; Zbl 1249.11011); translation in Sib. Adv. Math. 17, No. 4, 268--290 (2007)]. For the entire collection see [Zbl 1467.39001].