In this paper we study algebraic and differential relations for matrix valued orthogonal polynomials (MVOPs) defined on $\mathbb{R}$. Using recent results by Casper and Yakimov, we investigate MVOPs with respect to a matrix weight of the form $W(x)=e^{-v(x)}e^{xA} T e^{xA^\ast}$, where $v$ is a scalar polynomial of even degree with positive leading coefficient and $A$ and $T$ are constant matrices. We obtain ladder operators, discrete string equations for the recurrence coefficients and multi-time Toda equations for deformations with respect to parameters in the weight, and we show that the Lie algebra generated by the ladder operators is finite dimensional. Hermite-type matrix valued weights are studied in detail: in this case the weight is characterized by the ladder operators, and the Lie algebra generated by them can be extended to a Lie algebra that is isomorphic to the standard Harmonic oscillator algebra. Freud-type matrix weights are also discussed. Finally, we establish the link between these ladder relations and those considered previously by A. Duran and M. Ismail.

中文翻译：

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一类矩阵值正交多项式的梯形关系

在本文中，我们研究了定义在 $\mathbb{R}$ 上的矩阵值正交多项式 (MVOP) 的代数和微分关系。使用 Casper 和 Yakimov 最近的结果，我们研究了关于 $W(x)=e^{-v(x)}e^{xA} T e^{xA^\ast}$ 形式的矩阵权重的 MVOP ，其中 $v$ 是具有正领先系数的偶次标量多项式，$A$ 和 $T$ 是常数矩阵。我们获得了阶梯算子、递推系数的离散串方程和权重参数变形的多时间 Toda 方程，并且我们表明由阶梯算子生成的李代数是有限维的。详细研究了 Hermite 型矩阵值权重：在这种情况下，权重由阶梯算子表征，并且由它们生成的李代数可以扩展为与标准谐波振荡器代数同构的李代数。还讨论了弗洛伊德型矩阵权重。最后，我们建立了这些阶梯关系与 A. Duran 和 M. Ismail 之前考虑的那些关系之间的联系。