Solutions of inhomogeneous linear difference equations using Green's functions
S. R. Mane
TL;DR
This work extends Green's-function methods to inhomogeneous linear difference equations with variable coefficients by formulating retarded and advanced Green's functions $G_r(n,m)$ and $G_a(n,m)$. The particular solution is constructed as a convolution-like sum with the inhomogeneity $r(n)$, and the full solution combines a homogeneous part with these Green's-function-based contributions. The paper also connects discrete Casoratian determinants to the continuous Wronskian framework, demonstrates consistency with known constant-coefficient results, and illustrates the method with three detailed examples, including a variable-coefficient case. Overall, the approach provides a general, transferable framework for solving a broad class of inhomogeneous linear difference equations via Green's functions.
Abstract
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.