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

Abstract

We study a new family of orthogonal polynomials, here called big -1 Jacobi polynomials, which satisfy (apart from a 3-term recur- rence 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 orthogo- nal on the union of two symmetric intervals of the real axis. We show that the big -1 Jacobi polynomials can be obtained from the 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. The novelty lies in the fact that L is a differential-difference operator of special type. Namely, L is a linear operator which is of first order in the derivative operator @x and contains also the reflection operator R which acts as Rf(x) = f(−x). Roughly speaking, one can say that L belongs to the class of Dunkl operators (8) which contain both the operators @x and R. Nevertheless, the operator L differs from the standard Dunkl operators in a fundamental way. Indeed, L preserves the linear space of polynomials of any given maximal degree. This basic property allows to construct a complete system of polynomials Pn(x); n= 0;1;2;:::as eigenfunctions of the operator L. Guided by the q ! − 1 limit of the little q-Jacobi polynomials, we derived in (20) an explicit expression of the polynomials Pn(x) in terms of Gauss' hypergeometric functions. We also found explicitly the recurrence coefficients and s that the polynomials Pn(x) are orthogonal on the interval (−1;1) with a weight function related to the weight function of the generalized Jacobi polynomials (7). We also proved that they admit the Dunkl classical property (5) and further demonstrated that the operator L together with the multiplication operator x form a special case of the Askey-Wilson algebra AW(3) (21) corresponding to the parameter q = −1.