
SIGMA 8 (2012), 092, 20 pages arXiv:1212.0077
http://dx.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 AskeyWilson polynomials are orthogonal polynomials in
$x = \cos \theta$, which
are given as a terminating $_4\phi_3$ basic hypergeometric series.
The nonsymmetric AskeyWilson 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 AskeyWilson biorthogonality. These results include
discrete orthogonality relations. They can be considered as a classical analytic
study of the results for nonsymmetric
AskeyWilson polynomials which were previously obtained by affine Hecke
algebra techniques.
Key words:
AskeyWilson polynomials; orthogonality.
pdf (436 kb)
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