Nonisospectral deformations of noncommutative Laurent biorthogonal polynomials and matrix discrete Painlevé-type equations
Dan Dai, Xiaolu Yue
TL;DR
The paper addresses the link between noncommutative Laurent bi-OPs and matrix discrete Painlevé-type equations by applying nonisospectral deformations to derive a noncommutative mixed relativistic Toda lattice and its Lax pair. Through a stationary reduction, it obtains a matrix dP-type equation, with a weight-function-driven quasideterminant construction providing explicit solutions and justification for the reduction. The approach extends scalar discrete Painlevé II to the noncommutative/matrix setting and highlights the role of matrix-valued weights in generating quasideterminant solutions. Overall, the work advances noncommutative orthogonal polynomial theory, integrable lattice dynamics, and matrix Painlevé-type equations with potential applications in noncommutative geometry and matrix-valued random systems.
Abstract
In this paper, we establishes a connection between noncommutative Laurent biorthogonal polynomials (bi-OPs) and matrix discrete Painlevé (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.