非交换线性系的LU分解与雅可比多项式
LU-decomposition of a noncommutative linear system and Jacobi polynomials
摘要：
在本文中，我们得到的LU分解的非交换线性系的方程，在秩1的情况下，特征的图像的Lepowsky同态$U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$。这种LU分解可以转化为非常简单的矩阵恒等式，其中所涉及的矩阵的元素属于一类特殊的雅可比多项式。特别地，原始系统的L部分的每个条目用单个超球雅可比多项式表示。反过来，这些矩阵恒等式产生超球雅可比多项式之间的双正交关系。
英文摘要：
In this paper we obtain the LU-decomposition of a noncommutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$. This LU-decomposition can be transformed into very simple matrix identities, where the entries of the matrices involved belong to a special class of Jacobi polynomials. In particular, each entry of the L part of the original system is expressed in terms of a single ultraspherical Jacobi polynomial. In turns, these matrix identities yield a biorthogonality relation between the ultraspherical Jacobi polynomials.
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