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

SIGMA 9 (2013), 064, 14 pages      arXiv:1304.5866

Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System

Fethi Bouzeffour
Department of Mathematics, King Saudi University, College of Sciences, P.O. Box 2455 Riyadh 11451, Saudi Arabia

Received April 24, 2013, in final form October 16, 2013; Published online October 23, 2013; Misprints are corrected November 04, 2013

In this paper, we introduce a new differential-difference operator $T_\xi$ $(\xi \in \mathbb{R}^N)$ by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between $T_\xi$ and the directional derivative $\partial_\xi$. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this operator.

Key words: special functions; differential-difference operators; integral transforms.

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  1. Bouzeffour F., Special functions associated with complex reflection groups, Ramanujan J., to appear.
  2. Cherednik I., Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Int. Math. Res. Not. (1992), 171-180.
  3. Dunkl C.F., Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
  4. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  5. Dunkl C.F., Opdam E.M., Dunkl operators for complex reflection groups, Proc. London Math. Soc. 86 (2003), 70-108, math.RT/0108185.
  6. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  7. Heckman G.J., An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), 341-350.
  8. Heckman G.J., Dunkl operators, Astérisque 245 (1997), Exp.  No. 828, 4, 223-246.
  9. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
  10. Kober H., On fractional integrals and derivatives, Quart. J. Math., Oxford Ser. 11 (1940), 193-211.
  11. Koornwinder T.H., Bouzeffour F., Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials, Appl. Anal. 90 (2011), 731-746, arXiv:1006.1140.
  12. Luchko Y., Trujillo J.J., Caputo-type modification of the Erdélyi-Kober fractional derivative, Fract. Calc. Appl. Anal. 10 (2007), 249-267.
  13. Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
  14. Opdam E.M., Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), 333-373.
  15. Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1996.

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