Luc Vinet (Centre de Recherches Mathématiques, Université de Montréal, Canada)

The theory of Bannai-Ito polynomials

Abstract:
   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.
   Based on joint work with S. Tsujimoto (Kyoto University) and A. Zhedanov (Donetsk Institute for Science and Technology)