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

Orthogonal Basic Hypergeometric Laurent Polynomials

Mourad E. H. Ismailab, Dennis Stantonc

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 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.

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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Citation: Mourad E. H. Ismail, Dennis Stanton, “Orthogonal Basic Hypergeometric Laurent Polynomials”, SIGMA, 8 (2012), 092, 20 pp.

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\Bibitem{IsmSta12} \by Mourad~E.~H.~Ismail, Dennis~Stanton \paper Orthogonal Basic Hypergeometric Laurent Polynomials \jour SIGMA \yr 2012 \vol 8 \papernumber 092 \totalpages 20 \mathnet{http://mi.mathnet.ru/sigma769} \crossref{https://doi.org/10.3842/SIGMA.2012.092} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3007267} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000312436200001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84881537045} 

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This publication is cited in the following articles:
1. 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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