Dunkl shift operators and Bannai–Ito polynomials

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. We also present an anti-commutator AW(3) algebra corresponding to this operator. From the representations of this algebra, we derive the structure and recurrence relations of the BI polynomials. We introduce new orthogonal polynomials – referred to as the complementary BI polynomials – as an alternative q→−1 limit of the Askey–Wilson polynomials. These complementary BI polynomials lead to a new explicit expression for the BI polynomials in terms of the ordinary Wilson polynomials.