On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials

This paper continues recent investigations started in Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) into the structure of the set H_q,2n ³ {\mathcal{H}_{q,2n}^{\ge}} of all Hankel nonnegative definite sequences, (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}}, of complex q × q matrices and its important subclasses H_q,2n ^{3 ,e}{\mathcal{H}_{q,2n}^{\ge,{\rm e}}} and ${\mathcal{H}_{q,2n}^>}${\mathcal{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)_k=1ⁿ, (D_k)_k=0ⁿ]{(C_k)_{k=1}^n, (D_k)_{k=0}^n]} consisting of two sequences of complex q × q matrices was associated with an arbitrary sequence (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} of complex q × q matrices. The sequences belonging to each of the classes H_q,2n ³ , H_q,2n ^{3 ,e}{\mathcal{H}_{q,2n}^{\ge}, \mathcal{H}_{q,2n}^{\ge,{\rm e}}}, and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} were characterized in terms of their canonical Hankel parametrization (see, Dyukarev et al. in Complex 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)_k=1ⁿ, (D_k)_k=0ⁿ]{(C_k)_{k=1}^n, (D_k)_{k=0}^n]} of a sequence (s_j)_j=0²ⁿ ? H_q,2n ³ {(s_{j})_{j=0}^{2n} \in \mathcal{H}_{q,2n}^{\ge}}, we give a recursive construction of a monic right (resp. left) orthogonal system of matrix polynomials with respect to (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} (see Theorem 5.5). The matrices (C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be expressed in terms of an arbitrary monic right (resp. left) orthogonal system with respect to (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} (see Theorem 5.11). This result will be reformulated in terms of nonnegative Hermitian Borel measures on \mathbbR{\mathbb{R}}. In this way, integral representations for the matrices (C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be obtained (see Theorem 6.9). Starting from the monic orthogonal polynomials with respect to some classical probability distributions on \mathbbR{\mathbb{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.