Matrix Jacobi Biorthogonal Polynomials via Riemann-Hilbert problem

TL;DR

This paper develops a Riemann-Hilbert framework for matrix Jacobi orthogonal polynomials, deriving differential relations and non-Abelian Painlevé equations for their recurrence coefficients, advancing understanding of matrix orthogonal polynomial theory.

Contribution

It introduces a Riemann-Hilbert problem approach to matrix Jacobi polynomials and derives non-Abelian Painlevé equations for their recurrence coefficients, extending classical scalar results.

Findings

Derived differential relations for matrix orthogonal polynomials.

Established non-Abelian discrete Painlevé equations for recurrence coefficients.

Connected Riemann-Hilbert problems with matrix polynomial properties.

Abstract

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlev\'e d-PIV equations are obtained for the three term recurrence relation coefficients.