### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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.

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