Closed form solutions of linear difference equations in terms of symmetric products

TL;DR

This paper presents a method to derive closed form solutions for third order difference equations using second order solutions, enabling human-readable proofs and applications like sequence positivity verification.

Contribution

It extends previous recurrence solution methods by providing a systematic approach for higher order difference equations with verifiable proofs.

Findings

Closed form solutions for third order difference equations derived

Algorithm enables human-readable, certified proofs

Application demonstrated in sequence positivity problems

Abstract

In this paper we show how to find a closed form solution for third order difference operators in terms of solutions of second order operators. This work is an extension of previous results on finding closed form solutions of recurrence equations and a counterpart to existing results on differential equations. As motivation and application for this work, we discuss the problem of proving positivity of sequences given merely in terms of their defining recurrence relation. The main advantage of the present approach to earlier methods attacking the same problem is that our algorithm provides human-readable and verifiable, i.e., certified proofs.