
SIGMA, 2012, Volume 8, 092, 20 pages
(Mi sigma769)




This article is cited in 1 scientific paper (total in 1 paper)
Orthogonal Basic Hypergeometric Laurent Polynomials
Mourad E. H. Ismail^{ab}, Dennis Stanton^{c} ^{a} Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
^{b} Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
^{c} School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Abstract:
The Askey–Wilson polynomials are orthogonal polynomials in $x = \cos \theta$, which are given as a terminating $_4\phi_3$ basic hypergeometric series. The nonsymmetric Askey–Wilson polynomials are Laurent polynomials in $z=e^{i\theta}$, which are given as a sum of two terminating $_4\phi_3$'s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single $_4\phi_3$'s which are Laurent polynomials in $z$ are given, which imply the nonsymmetric Askey–Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for nonsymmetric Askey–Wilson polynomials which were previously obtained by affine Hecke algebra techniques.
Keywords:
Askey–Wilson polynomials; orthogonality
DOI:
https://doi.org/10.3842/SIGMA.2012.092
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MSC: 33D45 Received: August 4, 2012; in final form November 28, 2012; Published online December 1, 2012
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Mourad E. H. Ismail, Dennis Stanton, “Orthogonal Basic Hypergeometric Laurent Polynomials”, SIGMA, 8 (2012), 092, 20 pp.
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This publication is cited in the following articles:

Ismail M.E.H., Simeonov P., “Heine Representations and Monotonicity Properties of Determinants
and Pfaffians”, Constr. Approx., 41:2 (2015), 231–249

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