Matrix-valued Orthogonal Polynomials Related to (SU(2)$ \times$ SU(2), diag), II
Matrix-valued Orthogonal Polynomials Related to (SU(2)$ \times$ SU(2), diag), II
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与 (SU(2)$ imes$ SU(2), diag), II 相关的矩阵值正交多项式
DOI：
10.4171/prims/106
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发表时间：
2012
影响因子：
1.2
通讯作者：
P. Román
中科院分区：
文献类型：
--
作者：
E. Koelink;Maarten van Pruijssen;P. Román
在之前的论文中，我们通过研究 SU(2)\times SU(2) 上的矩阵值球函数，介绍了切比雪夫多项式的矩阵值类似物。特别是多项式的矩阵大小是任意大的。研究了矩阵值正交多项式和相应的权函数。特别是，我们计算权重的 LDU 分解，其中 L 的矩阵项以 Gegenbauer 多项式给出。模数矩阵值正交多项式 P_n 使用一阶和二阶矩阵值微分算子以 Tirao 矩阵值超几何函数的形式表示，其中 P_n 是其特征函数。根据这个结果，我们获得了多项式 P_n 满足的三项递推关系中的系数的显式公式。这些微分算子对于将 P_nL 的矩阵项表示为 Racah 和 Gegenbauer 多项式的乘积也至关重要。我们还通过考虑对应于 SU(2)\times SU(2) 的卡西米尔算子，提出了矩阵值微分算子的群论推导。
In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the P_n's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials P_n. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2)\times SU(2).