Rodica D. Costin
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a two-step recurrence relation, integral inter-relations, and quasi-orthogonality relations.
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Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2x2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence relation and their relation to matrix-valued q-hypergeometric series and the scalar-valued little q-Jacobi polynomials are presented. The study is based on a matrix-valued q-difference operator, which is a q-analogue of Tirao's matrix-valued hypergeometric differential operator.
Big $q$-Jacobi functions are eigenfunctions of a second order $q$-difference operator $L$. We study $L$ as an unbounded self-adjoint operator on an $L^2$-space of functions on $\mathbb R$ with a discrete measure. We describe explicitly the spectral decomposition of $L$ using an integral transform $\mathcal F$ with two different big $q$-Jacobi functions as a kernel, and we construct the inverse of $\mathcal F$.
Non-symmetric Jacobi polynomials of type $BC_{1}$ as vector-valued polynomials Part 1: spherical functions
本文通过向量值和矩阵值正交多项式研究 $BC_{1}$ 型非对称 Jacobi 多项式，揭示其与对称多项式及 Cherednik 算子的关系，并证明其与球面 $S^{2m+1}$ 上球面函数的等同性。
Non-symmetric Jacobi polynomials of type $BC_1$ as vector-valued polynomials Part 2: Shift operators
本文利用矩阵值正交多项式研究 $BC_1$ 型非对称 Jacobi 多项式，构造生成全部移位算子的四个基本微分–反射算子，并通过类 Harish-Chandra 同态建立其与球函数及 Lepowsky 同态的联系。
We write spectral decomposition of the hypergeometric differential operator on the contour $Re z=1/2$ (multiplicity of spectrum is 2). As a result, we obtain an integral transform that differs from the Jacobi (or Olevsky) transform. We also try to do a step towards vector-valued special functions and construct a 3F2 orthogonal basis in the space of functions having values in 2-dimensional space. This basis is lying in an analytic continuation of continuous dual Hahn polynomials with respect to number $n$ of a polynomial. In Addendum, we discuss a vector-value analog of the Meixner--Pollachek o…
Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as …
Asymptotics of matrix valued orthogonal polynomials on $[-1,1]$
本文利用 Riemann-Hilbert 表述和 Deift-Zhou 最速下降法，分析了权重含 Jacobi 标量因子的矩阵值正交多项式的大阶数渐近行为，并给出了复平面不同区域及三项递推关系系数的渐近展开。
In this paper we study moment sequences of matrix-valued measures on compact intervals. A complete parametrization of such sequences is obtained via a symmetric version of matricial canonical moments. Furthermore, distinguished extensions of finite moment sequences are characterized in this framework. The results are applied to the underlying matrix-valued measures, generalizing some results from the scalar theory of canonical moments.
We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulae for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.
In this paper we commence the study of discrete harmonic analysis associated with Jacobi orthogonal polynomials of order $(α,β)$. Particularly, we give the solution $W^{(α,β)}_t$, $t\ge 0$, and some properties of the heat equation related to the operator $J^{(α,β)}-I$, where $J^{(α,β)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator. These results will be a consequence of a much more general theorem concerning the solution of the heat equation for Jacobi matrices. In addition, we also prove the positivity of the operator $W^{(α,β)}_…
Consider a root system of type $BC_1$ on the real line $\mathbb R$ with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an $L^2$-space on $\mathbb R$ to a $L^2$-space of $\mathbb C^2$-valued functions on $\mathbb R^+$ with the Harish-Chandra measure $|c(\lam)|^{-2}d\lam$. By introducing a weight function of the form $\cosh^{-\sig}(t)\tanh^{2k} t$ on $\mathbb R$ we find an orthogonal basis for the $L^2$-space on $\mathbb R$ consisting of even and odd functions expressed in terms of the Jacobi polynomials (for each fixed $\sig$ and $k$). We find a R…
Probabilistic submeasures generalizing the classical (numerical) submeasures are introduced and discussed in connection with some classes of aggregation functions. A special attention is paid to triangular norm-based probabilistic submeasures and semi-copula-based probabilistic submeasures. Some algebraic properties of classes of such submeasures are also studied.
Orbits of Jacobi forms and Theta relations
本文提出Jacobi形式轨道的新方法，用于生成Jacobi theta函数恒等式，简化已知theta关系证明并推广Riemann theta关系。
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomials the operator $T$ can also be realized as a five-diagonal operator, hence leading to orthogonality relations for $2\times 2$-matrix-valued polynomials. These matrix-valued polynomials can be considere…
We consider semi-infinite Jacobi matrices corresponding to a point interaction for the discrete Schrödinger operator. Our goal is to find explicit expressions for the spectral measure, the resolvent and other spectral characteristics of such Jacobi matrices. It turns out that their spectral analysis leads to a new class of orthogonal polynomials generalizing the classical Chebyshev polynomials.
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