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

TL;DR

This paper introduces big -1 Jacobi polynomials, a new family of orthogonal polynomials derived as a limit of big q-Jacobi polynomials, with explicit formulas, orthogonality properties, and algebraic connections.

Contribution

The paper defines and analyzes big -1 Jacobi polynomials, including explicit hypergeometric representations, orthogonality on symmetric intervals, and their relation to Bannai-Ito and Askey-Wilson algebras.

Findings

Polynomials satisfy a Dunkl-type eigenvalue problem.

Explicit hypergeometric expression derived.

Orthogonal on two symmetric intervals.

Abstract

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 $q \to -1$. 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 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 $q \to -1$.