Solutions of inhomogeneous linear difference equations using Green's functions

We present a general formula for the particular solution of an inhomogeneous linear difference equation with variable coefficients. The answer is expressed as a weighted sum of fundamental solutions of the associated linear difference equation. This corresponds to an initial value problem in the case of linear differential equations. We remark that Green’s functions are naturally suited for solving such problems. This note presents a Green’s function formalism to solve an inhomogeneous linear difference equation with variable coefficients. Both the retarded and advanced Green’s functions are required, to obtain a complete solution. We independently confirm previous work for the case of linear difference equations with constant coefficients.