标题： 通过二次变换的矩阵值Hermite和Laguerre多项式 显示英文标题

标题： Matrix-Valued Hermite and Laguerre polynomials via Quadratic Transformation

摘要： 我们提出了经典Hermite-Laguerre二次对应关系到矩阵值设置的第一个系统扩展。 从一个Hermite类型的权矩阵W(x) = exp(-x^2) Z(x)开始，其中W(x) = W(-x)，变量替换y = x^2产生两个参数为alpha = -1/2和alpha = 1/2的Laguerre类型的权，并通过将相应的矩阵值正交多项式序列分解为偶数和奇数子序列来建立它们之间的显式关系。 我们证明了这种变换保持微分算子和Darboux变换，从而在Hermite侧和Laguerre侧之间建立了直接的结构联系，并为矩阵Bochner问题提供了新的构造工具。 具体的族——包括一个新的3x3示例和一个由块幂零矩阵构建的任意大小的族——说明了该理论，并提供了具有非平凡微分代数的矩阵值正交多项式的全新来源。 显示英文摘要

摘要： We present the first systematic extension of the classical Hermite-Laguerre quadratic correspondence to the matrix-valued setting. Starting from a Hermite-type weight matrix W(x) = exp(-x^2) Z(x) with W(x) = W(-x), the change of variables y = x^2 produces two Laguerre-type weights with parameters alpha = -1/2 and alpha = 1/2, and relates the corresponding sequences of matrix-valued orthogonal polynomials through an explicit decomposition into even and odd subsequences. We prove that this transformation preserves differential operators and Darboux transformations, thereby establishing a direct structural link between the Hermite and Laguerre sides and providing new constructive tools for the matrix Bochner problem. Concrete families - including a new 3x3 example and an arbitrary-size family built from block-nilpotent matrices - illustrate the theory and supply fresh sources of matrix-valued orthogonal polynomials endowed with non-trivial differential algebras.