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Askey-Wilson polynomials for root systems of type $BC$. (English) Zbl 0797.33014
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).
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 $R\sb 1=3D \{\pm 2 \varepsilon\sb j \}\sb{j=3D1,\dots,n}$, $R\sb 2=3D= \{\pm \varepsilon\sb i \pm \varepsilon\sb j\}\sb{1 \le i<j \le n}$. For the entire collection see [Zbl 0771.00045].

33D70Basic hypergeometric functions and integrals in several variables
33D80Connections of basic hypergeometric functions with groups, algebras and related topics
17B20Simple, semisimple, reductive Lie (super)algebras