On integrals for some class of ordinary difference equations admitting a Lax pair representation

TL;DR

This paper explores two classes of ordinary difference equations with Lax pair representations, describing their first integrals via special polynomials and revealing relationships between solutions of different equations based on parameters.

Contribution

It introduces a unified approach to describe first integrals of these classes and establishes equivalences and inclusion relations among their solution spaces.

Findings

First integrals expressed through special discrete polynomials.

Equivalence between difference equations of different classes with same parameters.

Organization of solution spaces in chains based on parameter sums.

Abstract

We consider two infinite classes of ordinary difference equations admitting Lax pair representation. Discrete equations in these classes are parameterized by two integers $k\geq 0$ and $s\geq k+1$. We describe the first integrals for these two classes in terms of special discrete polynomials. We show an equivalence of two difference equations belonged to different classes corresponding to the same pair $(k, s)$. We show that solution spaces $\mathcal{N}^k_s$ of different ordinary difference equations with fixed value of $s+k$ are organized in chain of inclusions.