Explicit and compact representations for the Green's function and the Solution of Linear Difference Equations with variable coefficients

TL;DR

This paper introduces explicit, compact representations for the Green's function and solutions of linear difference equations with variable coefficients, utilizing modified Leibniz formulas and Hessenbergian structures.

Contribution

It develops a novel Leibnizian and nested sum framework for representing Green's functions and fundamental solutions of difference equations with variable coefficients.

Findings

Explicit compact Green's function representation derived

Fundamental solutions expressed via banded Hessenbergian structures

Algorithms and software implementations demonstrate the results

Abstract

Leibniz' combinatorial formula for determinants is modified to establish a condensed and easily handled compact representation for Hessenbergians, referred to here as Leibnizian representation. Alongside, the elements of a fundamental solution set associated with linear difference equations with variable coefficients of order $p$ are explicitly represented by $p$ banded Hessenbergian solutions, built up solely of the variable coefficients. This yields banded Hessenbergian representations for the elements both of the product of companion matrices and of the determinant ratio formula of the one-sided Green's function (Green's function for short). Combining the above results, the elements of the foregoing notions are endowed with compact representations formulated here by Leibnizian and nested sum representations. We show that the elements of the fundamental solution set can be expressed in terms of the first banded Hessenbergian fundamental solution, called principal determinant function. We also show that the Green's function coincides with the principal determinant function, when both functions are restricted to a fairly large domain. These results yield, an explicit and compact representation of the Green's function restriction along with an explicit and compact solution representation of the previously stated type of difference equations in terms of the variable coefficients, the initial conditions and the forcing term. The equivalence of the Green's function solution representation and the well known single determinant solution representation is derived from first principles. Algorithms and automated software are employed to illustrate the main results of this paper.