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.

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 ...

The operator that intertwines between the $\mathbb{Z}_2$ - Dunkl and derivative is shown to have a realization in terms of oscillator operators one dimension. This observation rests on fact intertwining maps Hermite polynomials generalized polynomials.

In this paper, we proved a weighted Hardy–Rellich inequality for Dunkl operators based on the spherical h-harmonic decomposition theory of operators. Moreover, obtained explicit constant inequalities, which is optimal in some cases. Our results extend known inequalities.

In this note, a new proof for the positivity of Dunkl’s intertwining operator in the crystallographic case is given. It is based on an asymptotic relationship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, and on positivity results of S. Sahi for the Heckman-Opdam polynomials and their nonsymmetric counterparts. 2000 AMS Subject Classification: 33C52,...