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arXiv · 1011.1429

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

Luc Vinet·Alexei Zhedanov
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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 $q \to -1$.

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Luc Vinet, Alexei Zhedanov. 2010-11-25. A limit $q=-1$ for the big q-Jacobi polynomials. https://arxiv.org/abs/1011.1429

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