Computing Rational Generating Function of a Solution to the Initial Value Problem of Two-dimensional Difference Equation with Constant Coefficients

TL;DR

This paper introduces an algorithm to compute rational generating functions for solutions of two-dimensional difference equations, extending existing methods from one-dimensional cases and enabling applications in higher dimensions.

Contribution

The paper presents a novel algorithm for two-dimensional difference equations that reconstructs infinite initial data from finite input, facilitating higher-dimensional generalizations.

Findings

Algorithm successfully computes rational generating functions

Reconstruction of infinite initial data from finite data demonstrated

Potential for extension to higher-dimensional difference equations

Abstract

Algorithms for computing rational generating functions of solutions of one-dimensional difference equations are well-known and easy to implement. We propose an algorithm for computing rational generating functions of solutions of two-dimensional difference equations in terms of initial data of the corresponding initial value problems. The crucial part of the algorithm is the reconstruction of infinite one-dimensional initial data on the basis of finite input data. The proposed technique can be used for the development of similar algorithms in higher dimensions. We furnish examples of the implementation of the proposed algorithm.