# Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlev\'e IV

## TL;DR

This paper develops a Riemann-Hilbert framework for matrix biorthogonal Laguerre polynomials, deriving differential systems, eigenvalue problems, and non-Abelian discrete Painlevé IV equations, with an explicit example illustrating the theory.

## Contribution

It introduces a Riemann-Hilbert approach to analyze matrix Laguerre biorthogonal polynomials and connects them to eigenvalue problems and non-Abelian Painlevé equations.

## Key findings

- Derived differential systems for fundamental matrices.
- Established eigenvalue problems for matrix differential operators.
- Linked to non-Abelian discrete Painlevé IV equations.

## Abstract

In this paper the Riemann-Hilbert problem, with jump supported on a appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights ---which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann-Hilbert problem are derived. An explicit and general example is presented to illustrate the theoretical results of the work. Related matrix eigenvalue problems for second order matrix differential operators and non-Abelian extensions of a family of discrete Painlev\'e IV equations are discussed.