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Zbl 0797.33014
Koornwinder, Tom H.
Askey-Wilson polynomials for root systems of type $BC$. (English)
[A] Richards, Donald St. P. (ed.), Hypergeometric functions on domains of positivity, Jack polynomials, and applications. Proceedings of an AMS special session held March 22-23, 1991 in Tampa, FL, USA. Providence, RI: American Mathematical Society. Contemp. Math. 138, 189-204 (1992). ISBN 0-8218-5159-4

The author describes Macdonald's orthogonal polynomials associated with root systems, observes that for the root system $BC\sb 1$ these are a special case of the Askey-Wilson polynomials, and then finds a generalization of the Macdonald polynomials for $BC\sb n$ that introduces two additional parameters so that when $n=3D1$ these become the Askey- Wilson polynomials. These generalized Askey-Wilson polynomials are orthogonal with respect to the weight $$\prod\sb{\alpha \in R\sb 1} {(e\sp \alpha;q)\sb \infty \over (ae\sp{\alpha/2}, be\sp{\alpha/2}, ce\sp{\alpha/2}, de\sp{\alpha/2};q)\sb \infty} = 20 \prod {(e\sp \alpha; q)\sb \infty \over (te\sp \alpha;q)\sb \infty},$$ where $R\sb 1=3D \{\pm 2 \varepsilon\sb j \}\sb{j=3D1,\dots,n}$, $R\sb 2=3D= \{\pm \varepsilon\sb i \pm \varepsilon\sb j\}\sb{1 \le i<j \le n}$.
[D.M.Bressoud (St.Paul)]

MSC 2000:: *33D70 Basic hypergeometric functions and integrals in several variables
33D80 Connections with groups, algebras and related topics
17B20 Simple and semisimple Lie algebras

Keywords: Askey-Wilson polynomials

Cited in: Zbl 1132.33332 Zbl 1132.33334 Zbl 1080.33018 Zbl 1030.33010 Zbl 0941.33013 Zbl 0951.33010 Zbl 0874.33013 Zbl 0869.33012 Zbl 0858.39009

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