A limit $q=-1$ for the big $q$-Jacobi polynomials
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- by Luc Vinet and Alexei Zhedanov PDF
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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 $q \to -1$.References
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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
- MR Author ID: 178665
- ORCID: 0000-0001-6211-7907
- Alexei Zhedanov
- Affiliation: Institute for Physics and Engineering, R. Luxemburg str. 72, 83114 Donetsk, Ukraine
- MR Author ID: 234560
- 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
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 5491-5507
- MSC (2010): Primary 33C45, 33C47, 42C05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05539-5
- MathSciNet review: 2931336