A closed form to the general solution of linear difference equations with variable coefficients

TL;DR

This paper derives a closed-form solution for linear difference equations with variable coefficients by expressing Hessenbergian determinants as sums of elementary products, simplifying the solution process.

Contribution

It introduces a novel closed-form expression for Hessenbergian determinants, enabling explicit solutions for variable-coefficient linear difference equations.

Findings

Closed-form solution for linear difference equations with variable coefficients

Expressed Hessenbergian determinants as sums of elementary products

Applicable to N-order and ascending order difference equations

Abstract

The determinant of a lower Hessenberg matrix (Hessenbergian) is expressed as a sum of signed elementary products indexed by initial segments of nonnegative integers. A closed form alternative to the recurrence expression of Hessenbergians is thus obtained. This result further leads to a closed form of the general solution for regular order linear difference equations with variable coefficients, including equations of N-order and equations of ascending order.