
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 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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References
 Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of
Mathematics and its Applications, Vol. 71, Cambridge University Press,
Cambridge, 1999.
 Askey R., Wilson J., A set of orthogonal polynomials that generalize the
Racah coefficients or $6j$ symbols, SIAM J. Math. Anal.
10 (1979), 10081016.
 Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that
generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54
(1985), no. 319, iv+55 pages.
 Brezinski C., Biorthogonality and its applications to numerical analysis,
Monographs and Textbooks in Pure and Applied Mathematics, Vol. 156,
Marcel Dekker Inc., New York, 1992.
 Bultheel A., GonzalezVera P., Hendriksen E., Njastad O., Orthogonal rational
functions and continued fractions, in Special Functions 2000: Current
Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II
Math. Phys. Chem., Vol. 30, Kluwer Acad. Publ., Dordrecht, 2001, 87109.
 Cherednik I., Nonsymmetric Macdonald polynomials, Int. Math. Res.
Not. 1995 (1995), no. 10, 483515, qalg/9505029.
 Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of
Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University
Press, Cambridge, 2004.
 Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable,
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge
University Press, Cambridge, 2005.
 Ismail M.E.H., Masson D.R., Generalized orthogonality and continued fractions,
J. Approx. Theory 83 (1995), 140,
math.CA/9407213.
 Ismail M.E.H., Wilson J.A., Asymptotic and generating relations for the
$q$Jacobi and ${}_{4}\varphi_{3}$ polynomials, J. Approx.
Theory 36 (1982), 4354.
 Jones W.B., Thron W.J., Continued fractions, Encyclopedia of
Mathematics and its Applications, Vol. 11, AddisonWesley Publishing Co.,
Reading, Mass., 1980.
 Koekoek R., Swarttouw R.F., The Askeyscheme of hypergeometric orthogonal
polynomials and its $q$analogue, Report 9817, Faculty of Technical
Mathematics and Informatics, Delft University of Technology, 1998,
http://aw.twi.tudelft.nl/~koekoek/askey/.
 Koornwinder T.H., Bouzeffour F., Nonsymmetric AskeyWilson polynomials as
vectorvalued polynomials, Appl. Anal. 90 (2011), 731746,
arXiv:1006.1140.
 Lorentzen L., Waadeland H., Continued fractions with applications,
Studies in Computational Mathematics, Vol. 3, NorthHolland
Publishing Co., Amsterdam, 1992.
 Lorentzen L., Waadeland H., Continued fractions, Vol. 1, Convergence theory,
2nd ed., Atlantis Studies in Mathematics for Engineering and Science,
Atlantis Press, Paris, 2008.
 Macdonald I.G., Affine Hecke algebras and orthogonal polynomials,
Cambridge Tracts in Mathematics, Vol. 157, Cambridge University
Press, Cambridge, 2003.
 Noumi M., Stokman J.V., AskeyWilson polynomials: an affine Hecke algebra
approach, in Laredo Lectures on Orthogonal Polynomials and Special
Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci.
Publ., Hauppauge, NY, 2004, 111144, math.QA/0001033.
 Spiridonov V., Zhedanov A., Spectral transformation chains and some new
biorthogonal rational functions, Comm. Math. Phys. 210
(2000), 4983.
 Wilkinson J.H., The algebraic eigenvalue problem, Clarendon Press, Oxford,
1965.
 Wilson J.A., Private communication, 1980.
 Zhedanov A., Biorthogonal rational functions and the generalized eigenvalue
problem, J. Approx. Theory 101 (1999), 303329.

