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 | References | Similar Articles | Additional Information

Abstract: We construct a commutative algebra , generated by algebraically independent -difference operators acting on variables , which is diagonalized by the multivariable Askey-Wilson polynomials considered by Gasper and Rahman (2005). Iterating Sears' transformation formula, we show that the polynomials possess a certain duality between and . Analytic continuation allows us to obtain another commutative algebra , generated by algebraically independent difference operators acting on the discrete variables , which is also diagonalized by . This leads to a multivariable -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.

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© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.