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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 $B{C}_{1}$ these are a special case of the Askey-Wilson polynomials, and then finds a generalization of the Macdonald polynomials for $B{C}_{n}$ 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

$\prod _{\alpha \in {R}_{1}}\frac{{\left({e}^{\alpha };q\right)}_{\infty }}{{\left(a{e}^{\alpha /2},b{e}^{\alpha /2},c{e}^{\alpha /2},d{e}^{\alpha /2};q\right)}_{\infty }}=20\prod \frac{{\left({e}^{\alpha };q\right)}_{\infty }}{{\left(t{e}^{\alpha };q\right)}_{\infty }},$

where ${R}_{1}=3D{\left\{±2{\epsilon }_{j}\right\}}_{j=3D1,\cdots ,n}$, ${R}_{2}=3D={\left\{±{\epsilon }_{i}±{\epsilon }_{j}\right\}}_{1\le i.

##### MSC:
 33D70 Basic hypergeometric functions and integrals in several variables 33D80 Connections of basic hypergeometric functions with groups, algebras and related topics 17B20 Simple, semisimple, reductive Lie (super)algebras