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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Luc Vinet and Alexei Zhedanov
Journal: Trans. Amer. Math. Soc. 364 (2012), 5491-5507
MSC (2010): Primary 33C45, 33C47, 42C05
Published electronically: May 7, 2012
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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 (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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Additional Information

Luc Vinet
Affiliation: Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec, H3C 3J7 Canada

Alexei Zhedanov
Affiliation: Institute for Physics and Engineering, R. Luxemburg str. 72, 83114 Donetsk, Ukraine

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05539-5
PII: S 0002-9947(2012)05539-5
Keywords: Classical orthogonal polynomials, Jacobi polynomials, big $q$-Jacobi polynomials
Received by editor(s): November 29, 2010
Received by editor(s) in revised form: December 23, 2010, January 3, 2011, January 5, 2011, and January 8, 2011
Published electronically: May 7, 2012
Article copyright: © Copyright 2012 American Mathematical Society