Polynomial Solutions to the First Order Difference Equations in the Bivariate Difference Field

TL;DR

This paper introduces an algorithm to find all polynomial solutions to first order difference equations within a bivariate difference field, providing bounds on solution degrees to facilitate computation.

Contribution

It presents a novel algorithm for solving polynomial solutions of first order difference equations in bivariate difference fields, including degree bounds for solutions.

Findings

Algorithm successfully finds polynomial solutions

Provides degree bounds for solutions

Enables systematic computation of solutions

Abstract

The bivariate difference filed $(\mathbb{F}(\alpha, \beta), \sigma)$ provides an algebraic framework for a sequence satisfying a recurrence of order two and it could transform the summation involving a sequence satisfying a recurrence of order two into the first order difference equations in the bivariate difference field. Based on it, we present an algorithm for finding all the polynomial solutions of such equations in the bivariate difference field, and show an upper bound on the degree for polynomial solutions which is sufficient to compute polynomial solution by using the undetermined method.