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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 1 these are a special case of the Askey-Wilson polynomials, and then finds a generalization of the Macdonald polynomials for BC 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

αR 1 (e α ;q) (ae α/2 ,be α/2 ,ce α/2 ,de α/2 ;q) =20(e α ;q) (te α ;q) ,

where R 1 =3D{±2ε j } j=3D1,,n , R 2 =3D={±ε i ±ε j } 1i<jn .


MSC:
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