We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little q-Jacobi polynomials in the limit q = −1. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q = −1.

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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.

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Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-Pöschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.