Uniqueness Criterion and Cramer’s Rule for Implicit Higher Order Linear Difference Equations Over $$\mathbf {Z}$$

Abstract

We obtain uniqueness criterion for integer solutions of implicit higher order difference equations which extends the result of Berestovskii and Nikonorov. This criterion is specified for an implicit third order difference equation. We also obtain existence and uniqueness theorems for a solution of an implicit higher order nonhomogeneous difference equation over the ring of p-adic integers and over the ring \(\mathbf {Z}\). The possibility to obtain this solution by using the Cramer’s rule is established. We also give the explicit form for this solution. These results generalize corresponding results for implicit first and second order nonhomogeneous difference equations.