Related papers: Dunkl shift operators and Bannai-Ito polynomials
Yang-Baxter R operators symmetric with respect to the orthogonal and symplectic algebras are considered in an uniform way. Explicit forms for the spinorial and metaplectic R operators are obtained. L operators, obeying the RLL relation with…
Using a similarity transformation that maps the Calogero model into $N$ decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions. The…
A particular case of [07] was generalized from contractions to polynomially bounded operators in [G19]. Namely, it is proved in [G19] that if the unitary asymptote of a polynomially bounded operator $T$ contains the bilateral shift of…
Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the…
In this paper, we introduce a new differential-difference operator $T_\xi$ $(\xi \in \mathbb{R}^N)$ by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute…
The goal of this paper is to present a Dunkl-Gamma type operator with the help of two-variable Hermite polynomials and to derive its approximating properties via the classical modulus of continuity, second modulus of continuity and Peetre's…
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…
We introduce and analyze a novel class of binary operations on finite-dimensional vector spaces over a field K, defined by second-order multilinear expressions with linear shifts. These operations generate polynomials whose degree increases…
This note is devoted to the intertwining operator in the one--dimensional trigonometric Dunkl setting. We obtain a simple integral expression of this operator and deduce its positivity.
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…
We obtain the most general type B 3-fold supersymmetry by solving directly the intertwining relation. We then show that it is a necessary and sufficient condition for a second-order linear differential operator to have three linearly…
In this note, we express explicitly the Dunkl kernel and generalized Bessel functions of type $A_{n-1}$ by the Humbert's function $\Phi_{2}^{(n)}$, with one variable specified. The obtained formulas lead to a new proof of Xu's integral…
There is a commutative algebra of differential-difference operators, acting on polynomials on R_2, associated with the reflection group B2. This paper presents an integral transform which intertwines this algebra, allowing one free…
Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in $L^2(\mathbb R_+)$. These self-adjoint operators generalise the explicitly diagonalisable operator with the…
In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to a measure supported on the nonnegative integers. We show that the dual families are closely related to certain difference operators…
Quite some years ago, Oleg Chalykh has built a nice theory from the observation that the Macdonald polynomial reduces at $t=q^{-m}$ to a sum over permutations of simpler polynomials called Baker-Akhiezer functions, which can be…
Representations by linear integral operators on $L_p$ spaces over measure spaces are investigated for the polynomial covariance type commutation relations and more general two-sided generalizations of covariance commutation relations…
We introduce the -1 dual Hahn polynomials through an appropriate $q \to -1$ limit of the dual q-Hahn polynomials. These polynomials are orthogonal on a finite set of discrete points on the real axis, but in contrast to the classical…
In the present paper, we propose to give an extension to the context of Dunkl theory of the notion of translation and in connection with this a corresponding extension of Taylor's formula. More precisely, we prove some properties and…
Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on $\b R^N$. The definition and properties of these…