Dunkl shift operators and Bannai-Ito polynomials

@article{Tsujimoto2011DunklSO,
  title={Dunkl shift operators and Bannai-Ito polynomials},
  author={Satoshi Tsujimoto and Luc Vinet and Alexei S. Zhedanov},
  journal={arXiv: Classical Analysis and ODEs},
  year={2011}
}
We consider the most general Dunkl shift operator $L$ with the following properties: (i) $L$ is of first order in the shift operator and involves reflections; (ii) $L$ preserves the space of polynomials of a given degree; (iii) $L$ is potentially self-adjoint. We show that under these conditions, the operator $L$ has eigenfunctions which coincide with the Bannai-Ito polynomials. We construct a polynomial basis which is lower-triangular and two-diagonal with respect to the action of the operator… 
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