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Askey-Wilson polynomials: an affine Hecke algebraic approach | | Masatoshi Noumi
; Jasper V. Stokman
; | | Date: |
6 Dec 1999 | | Subject: | Quantum Algebra; Classical Analysis and ODEs; Representation Theory MSC-class: 33D45, 33D80 | math.QA math.CA math.RT | | Abstract: | We study Askey-Wilson type polynomials using representation theory of the double affine Hecke algebra. In particular, we prove bi-orthogonality relations for non-symmetric and anti-symmetric Askey-Wilson polynomials with respect to a complex measure. We give duality properties of the non-symmetric Askey-Wilson polynomials, and we show how the non-symmetric Askey-Wilson polynomials can be created from Sahi’s intertwiners. The diagonal terms associated to the bi-orthogonality relations (which replace the notion of quadratic norm evaluations for orthogonal polynomials) are expressed in terms of residues of the complex weight function using intertwining properties of the non-symmetric Askey-Wilson transform under the action of the double affine Hecke algebra. We evaluate the constant term, which is essentially the well-known Askey-Wilson integral, using shift operators. We furthermore show how these results reduce to well-known properties of the symmetric Askey-Wilson polynomials, as were originally derived by Askey and Wilson using basic hypergeometric series theory. | | Source: | arXiv, math.QA/0001033 | | Services: | Forum | Review | PDF | Favorites | |
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