  The renormalization method based on the Taylor expansion for asymptotic
analysis of differential equations is generalized to difference equations. The
proposed renormalization method is based on the Newton-Maclaurin expansion.
Several basic theorems on the renormalization method are proven. Some
interesting applications are given, including asymptotic solutions of quantum
anharmonic oscillator and discrete boundary layer, the reductions and invariant
manifolds of some discrete dynamics systems. Furthermore, the homotopy
renormalization method based on the Newton-Maclaurin expansion is proposed and
applied to those difference equations including no a small parameter.
