About several classes of bi-orthogonal polynomials and discrete integrable systems

TL;DR

This paper introduces special bi-orthogonal polynomials to derive new discrete integrable systems and algorithms, extending classical equations like the Toda and QQD schemes with associated Lax pairs.

Contribution

It develops novel bi-orthogonal polynomial-based methods to extend known discrete integrable systems and provides their Lax pairs, enriching the theory of discrete integrable equations.

Findings

Derived the discrete hungry quotient-difference (dhQD) algorithm.

Constructed extended systems for higher analogue discrete-time Toda and QQD schemes.

Provided Lax pairs for the new systems.

Abstract

By introducing some special bi-orthogonal polynomials, we derive the so-called discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka-Volterra (QD-type dhLV) system, together with their Lax pairs. These two known equations can be regarded as extensions of the QD algorithm. When this idea is applied to a higher analogue of the discrete-time Toda (HADT) equation and the quotient-quotient-difference (QQD) scheme proposed by Spicer, Nijhoff and van der Kamp, two extended systems are constructed. We call these systems the hungry forms of the higher analogue discrete-time Toda (hHADT) equation and the quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.