Implicit linear difference equations over a non-Archi-medean ring

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ for each initial value. It is interesting consider over ring, because case of a ring significantly different from one. The previous results on equations rings mostly concern integers and low order equations. In present article high some other classes rings, particularly, polynomials, are studied. To study integer idea considering p-adic completion with respect non-Archimedean valuation was useful. find solution such polynomials it naturally same construction this ring: formal power series valuation. both particular cases valuations fields: Laurent rational numbers respectively. arbitrary characteristic zero sufficient conditions uniqueness existence formulated. formula unique given, form sum series, converging Difference corresponds infinite system proved that in solution, be found using Cramer rules. Also facilitating finding polynomial given.