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

SIGMA 9 (2013), 064, 14 pages      arXiv:1304.5866      https://doi.org/10.3842/SIGMA.2013.064

### 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

Abstract
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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