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首页> 外文期刊>Complex analysis and operator theory >On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials
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On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials

机译：关于Hankel非负定序，规范的Hankel参数化和正交矩阵多项式

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This paper continues recent investigations started inDyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) into the structure of the set H≥q,2n of all Hankel nonnegative definite sequences, (s _j )2n j=0, of complex q × q matrices and its important subclasses H≥,e q,2n and H> q,2n of all Hankel nonnegative definite extendable sequences and of all Hankel positive definite sequences, respectively. These classes of sequences arise quite naturally in the framework of matrix versions of the truncated Hamburger moment problem. In Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) a canonical Hankel parametrization [(C_k )nk =1, (D_k )nk =0] consisting of two sequences of complex q × q matrices was associated with an arbitrary sequence (s _j )2n j=0 of complex q × q matrices. The sequences belonging to each of the classesH≥q,2n,H≥,e q,2n, andH> q,2n were characterized in terms of their canonical Hankel parametrization (see, Dyukarev et al. inComplex anal oper theory 3(4):759–834, 2009; Proposition 2.30). In this paper, we will study further aspects of the canonical Hankel parametrization. Using the canonical Hankel parametrization [(C_k )nk =1, (D_k )nk =0] of a sequence (s _j )2n j=0 ∈ H≥q,2n,we give a recursive construction of a monic right (resp. left) orthogonal system of matrix polynomials with respect to (s _j )2n j=0 (see Theorem 5.5). The matrices [(C_k )nk =1, (D_k )nk =0] will be expressed in terms of an arbitrary monic right (resp. left) orthogonal system with respect to (s j )2n j=0 (see Theorem 5.11). This result will be reformulated in terms of nonnegative Hermitian Borel measures on R. In this way, integral representations for the matrices [(C_k )nk =1, (D_k )nk=0] will be obtained (see Theorem 6.9). Starting from the monic orthogonal polynomials with respect to some classical probability distributions on R, Theorem 6.9 is used to compute the canonical Hankel parametrization of their moment sequences. Moreover, we discuss important number sequences from enumerative combinatorics using the canonical Hankel parametrization.
机译：本文继续了最近从Dyukarev等人开始的调查。 （复杂的肛门运算理论3（4）：759–834，2009）进入所有Hankel非负定序序列（s _j）2n j = 0的所有q×q矩阵和的H≥q，2n集的结构其所有Hankel非负定可扩展序列和所有Hankel正定序列的重要子类H≥，eq，2n和H> q，2n。这些类别的序列在截断的汉堡包矩问题的矩阵形式的框架中非常自然地出现。在Dyukarev等人。 （复杂肛门运算理论3（4）：759–834，2009年），由两个复杂q×q矩阵序列组成的规范汉克尔参数化[（C_k）nk = 1，（D_k）nk = 0]与任意复q×q矩阵的序列（s _j）2n j = 0。属于每个H≥q，2n，H≥，eq，2n和H> q，2n类别的序列根据其规范的汉克尔参数化来表征（请参阅Dyukarev等人在复杂的肛门手术理论3（4）中： 759–834，2009;建议2.30）。在本文中，我们将研究规范汉克尔参数化的其他方面。使用序列（s _j）2n j = 0∈H≥q，2n的规范汉克尔参数化[（C_k）nk = 1，（D_k）nk = 0]，我们给出一元权利（resp。左）关于（s_j）2n j = 0的矩阵多项式的正交系统（见定理5.5）。矩阵[（C_k）nk = 1，（D_k）nk = 0]将相对于（s j）2n j = 0（见定理5.11）以任意一元右（左）正交系统表示。将根据R上的非负Hermitian Borel度量来重新表示此结果。这样，将获得矩阵[（C_k）nk = 1，（D_k）nk = 0]的积分表示（请参见定理6.9）。从关于R上一些经典概率分布的单项​​正交多项式开始，定理6.9用于计算其矩序列的规范汉克尔参数化。此外，我们使用典型的汉克尔参数化讨论了来自枚举组合的重要数列。

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