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Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials. (English) Zbl 1216.33040
Summary: Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a \(2 \times 2\) matrix-valued operator. As a new result made possible by this approach, we obtain the positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little \(q\)-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey-Wilson and of the little \(q\)-Jacobi case.

MSC:
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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