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This paper introduces a family of Askey-Wilson type orthogonal polynomials in n variables associated with a root system of type BC n . The family depends, apart from q, on 5 parameters. For n = 1 it specializes to the four-parameter family of one-variable Askey-Wilson polynomials. For any n it contains Macdonald's two three-parameter families of orthogonal polynomials associated with a root system of type BC n as special cases.1991 Mathematics Subject Classification. Primary 33D45, 33D80, 17B20. Key words and phrases. Askey-Wilson polynomials, Macdonald's orthogonal polynomials associated with root systems, root system of type BC. CWI,

We study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4F3 hypergeometric functions and they satisfy second order differential equations. They include Krall’s Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way.

On the SU(2) quantum group the notion of (zonal) spherical element is generalized by considering left and right invariance in the infinitesimal sense with respect to twisted primitive elements of the sl(2) quantized universal enveloping algebra. The resulting spherical elements belonging to irreducible representations of quantum SU(2) turn out to be expressible as a two-parameter family of Askey-Wilson polynomials. For a related basis change of the representation space a matrix of dual q-Krawtchouk polynomials is obtained. Big and little q-Jacobi polynomials are obtained as limits of Askey-Wilson polynomials.