In this paper we use the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of non-Abelian Hermite biorthogonal polynomials in the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coefficients first order matrix polynomials, is given. All these is applied to the discussion of possible scenarios leading to eigenvalue problems for second order linear differential operators with matrix eigenvalues. Nonlinear matrix difference equations are discussed next. Firstly, for the general Hermite situation a general non linear relation (non trivial because the non commutativity features of the setting) for the recursion coefficients is gotten. In the next case of higher difficulty, degree two polynomials are allowed in the Pearson equation, but the discussion is simplified by considering only a left Pearson equation. In the case, the support of the measure is on an appropriate branch of an hyperbola. The recursion coefficients are shown to fulfill a non-Abelian extension of the alternate discrete Painlev\'e I equation. Finally, a discussion is given for the case of degree three polynomials as coefficients in the left Pearson equation characterizing the matrix of weights. However, for simplicity only odd polynomials are allowed. In this case, a new and more general matrix extension of the discrete Painlev\'e I equation is found.

中文翻译：

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矩阵双正交多项式：特征值问题和非阿贝尔离散 Painlevé 方程

在本文中，我们使用 Riemann-Hilbert 问题，在复平面中的适当曲线上支持跳跃，用于矩阵双正交多项式，并将其应用于寻找正交多项式及其第二类函数的 Sylvester 微分方程组。为此，权重矩阵的 Sylvester 型微分 Pearson 方程被证明是有用的。给出了几个应用程序，按复杂性增加的顺序。首先，给出了对实线中的非阿贝尔 Hermite 双正交多项式的一般讨论，这些多项式的权重矩阵是具有系数一阶矩阵多项式的 Sylvester 型 Pearson 方程的解。所有这些都应用于讨论导致具有矩阵特征值的二阶线性微分算子的特征值问题的可能场景。接下来讨论非线性矩阵差分方程。首先，对于一般 Hermite 情况，得到递归系数的一般非线性关系（非平凡，因为设置的非交换性特征）。在下一个难度更高的情况下，Pearson 方程中允许使用二阶多项式，但通过仅考虑左 Pearson 方程来简化讨论。在这种情况下，度量的支持位于双曲线的适当分支上。递归系数显示为满足交替离散 Painlev\'e I 方程的非阿贝尔扩展。最后，讨论了三次多项式作为表征权重矩阵的左皮尔逊方程中的系数的情况。然而，为简单起见，只允许奇数多项式。在这种情况下，发现了离散 Painlev\'e I 方程的一个新的和更一般的矩阵扩展。