Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

Details Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

2010-06-06

arxiv.org/abs/1006.1140v2

Mathematics

Classical Analysis and ODEs

16 pages. Dedicated to Paul Butzer on the occasion of his 80th birthday. v2: minor correction in (3.8). Appeared in Applicable

Scientific paper

10.1080/00036811.2010.502117

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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