Dunkl shift operators and Bannai-Ito polynomials

Understand

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 $L$.
• This allows to express the BI polynomials explicitly.