A limit 𝑞=-1 for the big 𝑞-Jacobi polynomials

We study a new family of “classical” orthogonal polynomials, here called big − 1 -1 Jacobi polynomials, which satisfy (apart from a 3 3 -term recurrence relation) an eigenvalue problem with differential operators of Dunkl type. These polynomials can be obtained from the big q q -Jacobi polynomials in the limit q → − 1 q → -1 . An explicit expression of these polynomials in terms of Gauss’ hypergeometric functions is found. The big − 1 -1 Jacobi polynomials are orthogonal on the union of two symmetric intervals of the real axis. We show that the big − 1 -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 q → -1 .