Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Special Issue on Dunkl Operators and Related Topics

The Guest Editors for this special issue are

Charles Dunkl (University of Virginia, USA)
Peter Forrester (University of Melbourne, Australia)
Marcel de Jeu (Leiden University, the Netherlands)
Margit Rösler (Technische Universität Clausthal, Germany)
Yuan Xu (University of Oregon, USA)

The original Dunkl operators and the associated Laplacian were defined by the undersigned in papers published in 1989 and 1988 respectively. The guiding idea was to use finite reflection groups to provide structure for multi-variable analysis. Twenty years later this special issue demonstrates that these methods are useful in harmonic analysis, integral transforms, probabilistic processes, special functions, quantum-mechanical integrable systems, differential equations, algebras associated to Coxeter groups, some aspects of representation theory and other topics. We thank the authors and the referees for their work and interest in the special issue. I gratefully acknowledge the efforts and valuable contributions of my fellow editors.

Charles F. Dunkl        

Papers in this Issue:

Dunkl Operators and Canonical Invariants of Reflection Groups
Arkady Berenstein and Yurii Burman
SIGMA 5 (2009), 057, 18 pages   [ abs   pdf   ps ]
Hilbert Transforms Associated with Dunkl-Hermite Polynomials
Néjib Ben Salem and Taha Samaali
SIGMA 5 (2009), 037, 17 pages   [ abs   pdf   ps ]
Nonlocal Operational Calculi for Dunkl Operators
Ivan H. Dimovski and Valentin Z. Hristov
SIGMA 5 (2009), 030, 16 pages   [ abs   pdf   ps ]
Inversion Formulas for the Spherical Radon-Dunkl Transform
Zhongkai Li and Futao Song
SIGMA 5 (2009), 025, 15 pages   [ abs   pdf   ps ]
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
Chokri Abdelkefi, Jean-Philippe Anker, Feriel Sassi and Mohamed Sifi
SIGMA 5 (2009), 019, 15 pages   [ abs   pdf   ps ]
Imaginary Powers of the Dunkl Harmonic Oscillator
Adam Nowak and Krzysztof Stempak
SIGMA 5 (2009), 016, 12 pages   [ abs   pdf   ps ]
Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type
Ta Khongsap and Weiqiang Wang
SIGMA 5 (2009), 012, 27 pages   [ abs   pdf   ps ]
The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials
Saburo Kakei, Michitomo Nishizawa, Yoshihisa Saito and Yoshihiro Takeyama
SIGMA 5 (2009), 010, 12 pages   [ abs   pdf   ps ]
An Alternative Definition of the Hermite Polynomials Related to the Dunkl Laplacian
Hendrik De Bie
SIGMA 4 (2008), 093, 11 pages   [ abs   pdf   ps ]
Sonine Transform Associated to the Dunkl Kernel on the Real Line
Fethi Soltani
SIGMA 4 (2008), 092, 14 pages   [ abs   pdf   ps ]
External Ellipsoidal Harmonics for the Dunkl-Laplacian
Hans Volkmer
SIGMA 4 (2008), 091, 13 pages   [ abs   pdf   ps ]
Symmetries of Spin Calogero Models
Vincent Caudrelier and Nicolas Crampé
SIGMA 4 (2008), 090, 15 pages   [ abs   pdf   ps ]
A Probablistic Origin for a New Class of Bivariate Polynomials
Michael R. Hoare and Mizan Rahman
SIGMA 4 (2008), 089, 18 pages   [ abs   pdf   ps ]
Dunkl Hyperbolic Equations
Hatem Mejjaoli
SIGMA 4 (2008), 084, 22 pages   [ abs   pdf   ps ]
A Limit Relation for Dunkl-Bessel Functions of Type A and B
Margit Rösler and Michael Voit
SIGMA 4 (2008), 083,   9 pages   [ abs   pdf   ps ]
Some Orthogonal Polynomials in Four Variables
Charles F. Dunkl
SIGMA 4 (2008), 082,   9 pages   [ abs   pdf   ps ]
Liouville Theorem for Dunkl Polyharmonic Functions
Guangbin Ren and Liang Liu
SIGMA 4 (2008), 076,   6 pages   [ abs   pdf   ps ]
Generalized Bessel function of Type D
Nizar Demni
SIGMA 4 (2008), 075,   7 pages   [ abs   pdf   ps ]
First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes
Nizar Demni
SIGMA 4 (2008), 074, 14 pages   [ abs   pdf   ps ]
Inversion Formulas for the Dunkl Intertwining Operator and Its Dual on Spaces of Functions and Distributions
Khalifa Trimèche
SIGMA 4 (2008), 067, 22 pages   [ abs   pdf   ps ]

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