Nonisospectral deformations of noncommutative Laurent biorthogonal polynomials and matrix discrete Painlev\'e-type equations

TL;DR

This paper explores nonisospectral deformations of noncommutative Laurent biorthogonal polynomials, linking them to matrix discrete Painlevé equations, and demonstrates how these deformations lead to integrable systems with specific reductions.

Contribution

It introduces a novel connection between noncommutative Laurent biorthogonal polynomials and matrix discrete Painlevé equations via nonisospectral deformations and stationary reductions.

Findings

Derived a noncommutative nonisospectral mixed relativistic Toda lattice

Established a matrix dP-type equation from the Lax pair reduction

Reduced the matrix dP equation to the known scalar alternate dP II equation

Abstract

In this paper, we establishes a connection between noncommutative Laurent biorthogonal polynomials (bi-OPs) and matrix discrete Painlev\'e (dP) equations. We first apply nonisospectral deformations to noncommutative Laurent bi-OPs to obtain the noncommutative nonisospectral mixed relativistic Toda lattice and its Lax pair. Then, we perform a stationary reduction on this Lax pair to obtain a matrix dP-type equation. The validity of this reduction is demonstrated through a specific choice of weight function and the application of quasideterminant properties. In the scalar case, our matrix dP equation reduces to the known alternate dP II equation.