SIGMA 8 (2012), 092, 20 pages      arXiv:1212.0077      https://doi.org/10.3842/SIGMA.2012.092
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Orthogonal Basic Hypergeometric Laurent Polynomials

Mourad E.H. Ismail ^{a, b} and 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

Received August 04, 2012, in final form November 28, 2012; Published online December 01, 2012

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 non-symmetric 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 non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.

Key words: Askey-Wilson polynomials; orthogonality.

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