Matrix biorthogonal polynomials in the unit circle: Riemann-Hilbert problem and matrix discrete Painleve II system

TL;DR

This paper develops a Riemann-Hilbert framework for matrix biorthogonal polynomials on the unit circle, deriving new matrix discrete Painleve II systems and exploring their properties through differential and difference equations.

Contribution

It introduces a Riemann-Hilbert problem approach for matrix Szego polynomials, leading to novel matrix discrete Painleve II equations and their reductions.

Findings

Established a unique Riemann-Hilbert problem solution for matrix Szego polynomials.

Derived matrix discrete Painleve II equations for Verblunsky matrices.

Analyzed Fuchsian and non-Fuchsian cases with new nonlinear difference equations.

Abstract

Matrix Szego biorthogonal polynomials for quasi-definite matrices of measures are studied. For matrices of Holder weights a Riemann-Hilbert problem is uniquely solved in terms of the matrix Szego polynomials and its Cauchy transforms. The Riemann-Hilbert problem is given as an appropriate framework for the discussion of the Szego matrix and the associated Szego recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations which are required to be monodromy free. Linear ordinary differential equations for the matrix Szego polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non-Fuchsian type, are considered. For both cases a new matrix version of the discrete Painlev\'e II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painleve II systems presenting locality are discussed.