We investigate algebraic structure for the BN -type Calogero model by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis.

For a finite reflection group on R , the associated Dunkl operators are parametrized firstorder differential-difference operators which generalize the usual partial derivatives. They generate a commutative algebra which is – under weak assumptions – intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials. In this paper it is s...

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on RN . The Dunkl operators are denoted by Tj for 1 ≤ j ≤ N, and the Laplacian ∆κ = ∑j=1 T2 j . This paper finds the homogeneous harmonic polynomials annihilated by all Tj for j > 2. The structure constants with respect to the Gaussian and sphere inner...

Recent studies show that deformations in quantum mechanics are inevitable. In this contribution, we consider a relativistic mechanical differential equation the presence of Dunkl operator-based deformation and investigate solutions for two important problems three-dimensional spatial space. To end, after introducing mechanics, examine Dunkl–Klein–Gordon oscillator with Cartesian spherical coord...

In 1961, Bargmann introduced the classical Fock space $$\mathscr {F}(\mathbb {C}^{d})$$ and in 1984, Cholewinsky generalized {F}_{\alpha ,e}(\mathbb . These two spaces are aim of many works, have applications mathematics, physics, quantum mechanics. this work, we introduce study }(\mathbb associated to Dunkl operators $$T_{\alpha _{j}}$$ with $$\alpha _{j}>-1/2$$ for all $$j=1,\ldots ,d$$ This ...

This paper consists in a first study of the Hardy space H in the rational Dunkl setting. Following Uchiyama’s approach, we characterizee H atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H. These results are proved here in the one-dimensional case and in the product case.

In this paper we prove inversion formulas for the Dunkl intertwining operator Vk and for its dual tVk and we deduce the expression of the representing distributions of the inverse operators V −1 k and V −1 k , and we give some applications.

Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions jx1jã1 Ð Ð Ð jxdjãd on the unit sphere Sd 1 in Rd. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.