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Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

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arxiv 1006.1140 v3 pith:OOBC3TEI submitted 2010-06-06 math.CA math.QA

Tom H. Koornwinder , Fethi Bouzeffour This is my paper · ORCID
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keywords polynomialsnonsymmetricaskey-wilsonlaurentlittleoperatorpossibleq-jacobi
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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 2x2 matrix-valued operator. As a new result made possible by this approach we obtain 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.

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  1. Bispectrality of the sieved Jacobi polynomials

    math.CA 2025-01 conditional novelty 8.0 of 10

    The sieved Jacobi polynomials are shown to be eigenfunctions of new Dunkl-type differential operators, so they are bispectral.

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