Generalized backward shift operators on the ring ℤ[[x]], Cramer’s rule for infinite linear systems, and p-adic integers

Abstract

Let A be a generalized backward shift operator on ℤ[[x]] and f(x) be a formal power series with integer coefficients. A criterion for the existence of a solution of the linear equation (Ay)(x) + f(x) = y(x) in ℤ[[x]] is obtained. An explicit formula for its unique solution in ℤ[[x]] is found as well. The main results are based on using the p-adic topology on ℤ and on using a formal version of Cramer’s rule for solving infinite linear systems.