A LIMIT q =-1 FOR THE BIG q-JACOBI POLYNOMIALS

We study a new family of classical orthogonal polynomials, here called big - 1 Jacobi polynomials, which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with differential operators of Dunkl type. These polynomials can be obtained from the big q-Jacobi polynomials in the limit q -> -1. An explicit expression of these polynomials in terms of Gauss' hypergeometric functions is found. The big 1 Jacobi polynomials are orthogonal on the union of two symmetric intervals of the real axis. We show that the big - 1 Jacobi polynomials can be obtained from the (terminating) Bannai-Ito polynomials when the orthogonality support is extended to an infinite number of points. We further indicate that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q -> -1.