相关论文: Real-Root Preserving Differential Operator Represe…

Heckman-Polychronakos operators form a prominent family of commuting differential-difference operators defined in terms of the Dunkl operators $\mathcal D_i$ as $\mathcal P_m= \sum_{i=1}^N (x_i \mathcal D_i)^m$. They have been known since…

In this work, we develop a constructive method for deriving four structure relations and a fourth-order linear differential equation satisfied by Laguerre-Hahn orthogonal polynomial sequences. The method relies on a combination of structure…

By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie algebras, Luo and the second author (2013) obtained a large family of infinite-dimensional irreducible representations of the…

Let $R_+: = [ 0 , +\infty) $. Assume that $ n \times n$ ($ n \in \mathcal{N} $) matrix functions $P, Q $ and $ R $ are defined on the set $R_+ $, $P(x)$ is non-degenerate, $P(x)$ and $Q(x)$ are Hermitian matrices when $x \in R_+$ and the…

We study hyperbolic polynomials with nice symmetry and express them as the determinant of a Hermitian matrix with special structure. The goal of this paper is to answer a question posed by Chien and Nakazato in 2015. By properly modifying a…

In this paper, we show that the commonly used frame soft shrinkage operator, that maps a given vector ${\mathbf x} \in {\mathbb R}^{N}$ onto the vector ${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T} {\mathbf x}$, is already a proximity…

The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as…

It will be shown here that there are differential operators $E,F$ and $H=[E,F]$ for each $n\ge 1$, acting on Diagonal Harmonics, yielding that $DH_n$ is a representation of $sl[2]$ (see [3] Chapter 3). Our main effort here is to use $sl[2]$…

A polynomial transform is the multiplication of an input vector $x\in\C^n$ by a matrix $\PT_{b,\alpha}\in\C^{n\times n},$ whose $(k,\ell)$-th element is defined as $p_\ell(\alpha_k)$ for polynomials $p_\ell(x)\in\C[x]$ from a list…

In this paper, we study parameter deformations of matrix valued orthogonal polynomials (MVOPs). These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product…

Exceptional orthogonal Hermite and Laguerre polynomials have been linked to the k-step extension of harmonic and singular oscillators. The exceptional polynomials allow the existence of different supercharges from the Darboux-Crum and…

Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied.…

Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four…

In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper to the $q$-deformed case. A generalized…

We show that any scalar differential operator with a family of polyno- mials as its common eigenfunctions leads canonically to a matrix differen- tial operator with the same property. The construction of the correspond- ing family of matrix…

The Ruijsenaars-Schneider systems are `discrete' version of the Calogero-Moser (C-M) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e^{\pm\beta' p} (\beta' is a deformation parameter)…

We construct explicit differential operators on hermitian modular forms, extending methods developed for Siegel modular forms. These differential operators are closely related to the two-variable spherical pluriharmonic polynomials. We…

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for…

Integrability, algebraic structures and orthogonal basis of the Calogero model are studied by the quantum Lax and Dunkl operator formulations. The commutator algebra among operators including conserved operators and creation-annihilation…

We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald $P$-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our…