Nonsymmetric Askey–Wilson polynomials as vector-valued polynomials

Abstract

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 × 2 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.

Keywords:

• nonsymmetric Askey–Wilson polynomials
• vector-valued polynomials
• Dunkl–Cherednik operator
• positive-definite inner product
• nonsymmetric little q-Jacobi polynomials
• non-symmetric Jacobi polynomials

AMS Subject Classifications::

• 33D45
• 33D52
• 33C45