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

SIGMA 8 (2012), 092, 20 pages      arXiv:1212.0077
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

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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  1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  2. Askey R., Wilson J., A set of orthogonal polynomials that generalize the Racah coefficients or $6-j$ symbols, SIAM J. Math. Anal. 10 (1979), 1008-1016.
  3. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55 pages.
  4. 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.
  5. Bultheel A., Gonzalez-Vera 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, 87-109.
  6. Cherednik I., Nonsymmetric Macdonald polynomials, Int. Math. Res. Not. 1995 (1995), no. 10, 483-515, q-alg/9505029.
  7. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
  8. 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.
  9. Ismail M.E.H., Masson D.R., Generalized orthogonality and continued fractions, J. Approx. Theory 83 (1995), 1-40, math.CA/9407213.
  10. 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), 43-54.
  11. Jones W.B., Thron W.J., Continued fractions, Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980.
  12. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998,
  13. Koornwinder T.H., Bouzeffour F., Non-symmetric Askey-Wilson polynomials as vector-valued polynomials, Appl. Anal. 90 (2011), 731-746, arXiv:1006.1140.
  14. Lorentzen L., Waadeland H., Continued fractions with applications, Studies in Computational Mathematics, Vol. 3, North-Holland Publishing Co., Amsterdam, 1992.
  15. Lorentzen L., Waadeland H., Continued fractions, Vol. 1, Convergence theory, 2nd ed., Atlantis Studies in Mathematics for Engineering and Science, Atlantis Press, Paris, 2008.
  16. Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
  17. Noumi M., Stokman J.V., Askey-Wilson 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, 111-144, math.QA/0001033.
  18. Spiridonov V., Zhedanov A., Spectral transformation chains and some new biorthogonal rational functions, Comm. Math. Phys. 210 (2000), 49-83.
  19. Wilkinson J.H., The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965.
  20. Wilson J.A., Private communication, 1980.
  21. Zhedanov A., Biorthogonal rational functions and the generalized eigenvalue problem, J. Approx. Theory 101 (1999), 303-329.

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