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

Vinet, Luc; Zhedanov, Alexei

doi:10.1090/S0002-9947-2012-05539-5

A limit $q=-1$ for the big $q$-Jacobi polynomials
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Trans. Amer. Math. Soc. 364 (2012), 5491-5507 Request permission

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 (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 $q \to -1$.

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