Matrix Szegő biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szegő polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szegő matrix and the associated Szegő 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 that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szegő 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 Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.

中文翻译：

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Riemann-Hilbert问题和矩阵离散PainlevéII系统

研究了Hölder连续权准定矩阵的矩阵Szegő双正交多项式。黎曼-希尔伯特问题可以通过矩阵Szegő多项式及其Cauchy变换来唯一解决。给出了黎曼-希尔伯特问题作为讨论Szegő矩阵以及矩阵正交多项式及其Cauchy变换的相关Szegő递归关系的适当框架。研究了表征权重矩阵的Pearson型微分系统。这些是具有平凡单峰性的常微分方程的线性系统。推导了矩阵Szegő多项式的线性常微分方程及其Cauchy变换。展示了这些Pearson系统如何导致Verblunsky矩阵的非线性差分方程和两个示例，考虑了Fuchsian和非Fuchsian类型。对于这两种情况，都找到了用于Verblunsky矩阵的离散PainlevéII方程的新矩阵形式。讨论了这些呈现局部性的矩阵离散PainlevéII系统的约简。