满足高阶递归的矩阵微分方程与标量多项式
Matrix differential equations and scalar polynomials satisfying higher order recursions
摘要：
我们证明了任何以多项式族为公共本征函数的标量微分算子正则地导出具有相同性质的矩阵微分算子。[D1，D2，DV]中研究了矩阵值多项式的对应族的构造，但没有考虑以它们为公共本征函数的微分算子的存在性。这种对应关系只有一种方式，大多数矩阵值情况并不以这种方式出现。我们用几个例子来说明这个一般结构。在一些家庭的情况下，标量值多项式介绍[GH]，我们首先看看所有矩阵微分算子，共享这些共同的本征函数的代数，并揭示了一些现象，是新的矩阵值的情况下。
英文摘要：
We show that any scalar differential operator with a family of polyno- mials as its common eigenfunctions leads canonically to a matrix differen- tial operator with the same property. The construction of the correspond- ing family of matrix valued polynomials has been studied in [D1, D2, DV] but the existence of a differential operator having them as common eigen- functions had not been considered This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [GH] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.
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