Classical ellipsoidal and sphero-conal harmonics are polynomial solutions of the Laplace equation that can be expressed in terms of Lamé polynomials. Generalized ellipsoidal and sphero-conal harmonics are polynomial solutions of the more general Dunkl equation that can be expressed in terms of Stieltjes polynomials. Niven’s formula connecting ellipsoidal and sphero-conal harmonics is generalize...

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 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,...

Using a Bessel generalized translation, we obtain an analog of Titchmarsh's theorem for the Bessel transform for functions satisfying the Lipschitz condition in the space $mathrm{L}_{p,alpha}(mathbb{R}_{+})$, where $alpha>-frac{1}{2}$ and $1

Generalized translation operators for orthogonal expansions with respect to families of weight functions on the unit ball and on the standard simplex are studied. They are used to define convolution structures and modulus of smoothness for these regions, which are in turn used to characterize the best approximation by polynomials in the weighted Lp spaces. In one variable, this becomes the gene...

The goal of this paper is to present a Dunkl-Gamma type operator with the help generalization two-variable Hermite polynomials and derive its approximating properties via classical modulus continuity, second continuity Peetre’s K-functional.

we introduce a new concept of general $g$-$eta$-monotone operator generalizing the general $(h,eta)$-monotone operator cite{arvar2, arvar1}, general $h-$ monotone operator cite{xiahuang} in banach spaces, and also generalizing $g$-$eta$-monotone operator cite{zhang}, $(a, eta)$-monotone operator cite{verma2}, $a$-monotone operator cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}...