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Affine Dunkl processes of type ${\tilde{A}}_{1}$

François Chapon (2012)

Annales de l'I.H.P. Probabilités et statistiques

We introduce the analogue of Dunkl processes in the case of an affine root system of type ${\tilde{A}}_{1}$ . The construction of the affine Dunkl process is achieved by a skew-product decomposition by means of its radial part and a jump process on the affine Weyl group, where the radial part of the affine Dunkl process is given by a Gaussian process on the ultraspherical hypergroup $[0, 1]$ . We prove that the affine Dunkl process is a càdlàg Markov process as well as a local martingale, study its jumps, and give a martingale...

Affine Hecke algebras and orthogonal polynomials

I. G. MacDonald (1994/1995)

Séminaire Bourbaki

An analytic formula for the $A_{2}$ Jack polynomials.

Mangazeev, Vladimir V. (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

An explicit formula for symmetric polynomials related to the eigenfunctions of Calogero-Sutherland models.

Hallnäs, Martin (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Biorthogonal expansion of non-symmetric Jack functions.

Sahi, Siddhartha, Zhang, Genkai (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Discrete Laguerre functions and equilibrium conditions.

Schipp, Ferenc (2001)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

External ellipsoidal harmonics for the Dunkl-Laplacian.

Volkmer, Hans (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

First hitting time of the boundary of the Weyl chamber by radial Dunkl processes.

Demni, Nizar (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Generalized Bessel function of type $D$ .

Demni, Nizar (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Liouville theorem for Dunkl polyharmonic functions.

Ren, Guangbin, Liu, Liang (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Littlewood-Paley $g$ -function in the Dunkl analysis on $ℝ^{d}$ .

Soltani, Fethi (2005)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Orthogonal polynomials associated with root systems.

Macdonald, Ian G. (2000)

Séminaire Lotharingien de Combinatoire [electronic only]

Orthogonality within the families of $C$ -, $S$ -, and $E$ -functions of any compact semisimple Lie group.

Moody, Robert V., Patera, Jiri (2006)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Some orthogonal polynomials in four variables.

Dunkl, Charles F. (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

The rational qKZ equation and shifted non-symmetric Jack polynomials.

Kakei, Saburo, Nishizawa, Michitomo, Saito, Yoshihisa, Takeyama, Yoshihiro (2009)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Three results in Dunkl analysis

Béchir Amri, Jean-Philippe Anker, Mohamed Sifi (2010)

Colloquium Mathematicae

We first establish a geometric Paley-Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the $L^{p} \to L^{p}$ norm of Dunkl translations in dimension 1. Finally, we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.

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