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Bispectral commuting difference operators for multivariable Askey-Wilson polynomials


Author: Plamen Iliev
Journal: Trans. Amer. Math. Soc. 363 (2011), 1577-1598
MSC (2010): Primary 39A14, 33D50
DOI: https://doi.org/10.1090/S0002-9947-2010-05183-9
Published electronically: October 25, 2010
MathSciNet review: 2737278
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Abstract: We construct a commutative algebra $ \mathcal A_{z}$, generated by $ d$ algebraically independent $ q$-difference operators acting on variables $ z_1,z_2,\dots,z_d$, which is diagonalized by the multivariable Askey-Wilson polynomials $ P_n(z)$ considered by Gasper and Rahman (2005). Iterating Sears' $ {}_4\phi_3$ transformation formula, we show that the polynomials $ P_n(z)$ possess a certain duality between $ z$ and $ n$. Analytic continuation allows us to obtain another commutative algebra $ \mathcal A_{n}$, generated by $ d$ algebraically independent difference operators acting on the discrete variables $ n_1,n_2,\dots,n_d$, which is also diagonalized by $ P_n(z)$. This leads to a multivariable $ q$-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.


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Additional Information

Plamen Iliev
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332–0160
Email: iliev@math.gatech.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05183-9
Received by editor(s): July 20, 2009
Received by editor(s) in revised form: August 12, 2009, and August 14, 2009
Published electronically: October 25, 2010
Additional Notes: The author was supported in part by NSF Grant #0901092.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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