Stationary reduction method based on nonisospectral deformation of orthogonal polynomials, and discrete Painlev\'{e}-type equations

TL;DR

This paper introduces a novel stationary reduction method based on nonisospectral deformation of orthogonal polynomials to derive new discrete Painlevé-type equations and integrable difference systems, revealing connections with Toda hierarchies.

Contribution

It presents a new approach for deriving discrete Painlevé-type equations from various orthogonal polynomials, including novel classes and their solutions with Lax pairs.

Findings

Derived several new high order d-P-type equations.

Connected integrable difference systems to Toda hierarchies.

Found Pfaffian solutions for systems related to partial-skew orthogonality.

Abstract

In this work, we propose a new approach called ``stationary reduction method based on nonisospectral deformation of orthogonal polynomials" for deriving discrete Painlev\'{e}-type (d-P-type) equations. We apply this approach to (bi)orthogonal polynomials satisfying ordinary orthogonality, $(1,m)$-biorthogonality, generalized Laurent biorthogonality, Cauchy biorthogonality and partial-skew orthogonality. As a result, several seemingly novel classes of high order d-P-type equations or integrable difference systems with potential relations with new d-P-type equations, along with their particular solutions and respective Lax pairs, are derived. Notably, the derived integrable difference system related to the Cauchy biorthogonality is a stationary reduction of a nonisospectral generalization involving the first two flows of the Toda hierarchy of CKP type. Additionally, the integrable difference system related to the partial-skew orthogonality is associated with the nonisospectral Toda hierarchy of BKP type, and it is found to admit a solution expressed in terms of Pfaffians.