Askey-Wilson Functions and Quantum Groups

  • Jasper V. Stokman
Part of the Developments in Mathematics book series (DEVM, volume 13)


Eigenfunctions of the Askey-Wilson second order q-difference operator for 0 < q < 1 and |q| = 1 are constructed as formal matrix coefficients of the principal series representation of the quantized universal enveloping algebra U q (\(\mathfrak{s}\mathfrak{l}\)(2, ℂ)). The eigenfunctions are given in integral form. We show that for 0 < {tiq} < 1 the resulting eigenfunction can be rewritten as a very-well-poised 8ϕ7-series, and reduces for special parameter values to a natural elliptic analogue of the cosine kernel.


Meromorphic Function Quantum Group Expansion Formula Meromorphic Continuation Basic Hypergeometric Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Askey, R. and Wilson, J. A. (1985). Some basic hypergeometric polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 54(319).Google Scholar
  2. Cherednik, I. (1997). Difference Macdonald-Mehta conjecture. Internat. Math. Res. Notices, 10:449–467.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Faddeev, L. (2000). Modular double of a quantum group. In Conférence Moshé Flato 1999, Vol. I (Dijon), volume 21 of Math Phys. Stud., pages 149–156. Kluwer Acad. Publ., Dordrecht.Google Scholar
  4. Gasper, G. and Rahman, M. (1990). Basic hypergeometric series, volume 35 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  5. Ismail, M. and Rahman, M. (1991). The associated Askey-Wilson polynomials. Trans. Amer. Math. Soc., 328:201–237.CrossRefMathSciNetzbMATHGoogle Scholar
  6. Kharchev, S., Lebedev, D., and Semenov-Tian-Shansky, M. (2002). Unitary representations of u q(sl(2,R)), the modular double and the multiparticle q-deformed Toda chains. Comm. Math. Phys., 225(3):573–609.CrossRefMathSciNetzbMATHGoogle Scholar
  7. Koelink, E. (1996). Askey-Wilson polynomials and the quantum SU(2) group: survey and applications. Acta Appl. Math., 44(3):295–352.zbMATHMathSciNetGoogle Scholar
  8. Koelink, E. and Rosengren, H. (2002). Transmutation kernels for the little q-Jacobi function transform. Rocky Mountain J. Math., 32(2):703–738. Conference on Special Functions (Tempe, AZ, 2000).CrossRefMathSciNetzbMATHGoogle Scholar
  9. Koelink, E. and Stokman, J. V. (2001a). The Askey-Wilson function transform. Internat. Math. Res. Notices, 22:1203–1227.CrossRefMathSciNetGoogle Scholar
  10. Koelink, E. and Stokman, J. V. (2001b). Fourier transforms on the quantum SU(1, 1) group. Publ. Res. Inst. Math. Sci., 37(4):621–715. With an appendix by M. Rahman.MathSciNetzbMATHGoogle Scholar
  11. Koornwinder, T. H. (1984). Jacobi functions and analysis on noncompact semisimple Lie groups. In Special Functions: Group Theoretical Aspects and Applications, Math. Appl., pages 1–85. Reidel, Dordrecht.Google Scholar
  12. Koornwinder, T. H. (1993). Askey-Wilson polynomials as zonal spherical functions on the su(2) quantum group. SIAM J. Math. Anal., 24:795–813.CrossRefzbMATHMathSciNetGoogle Scholar
  13. Nishizawa, M. (2001). q-special functions with |q| = 1 and their application to discrete integrable systems. J. Phys. A, 34(48):10639–10646. Symmetries and Integrability of Difference Equations (Tokyo, 2000).CrossRefzbMATHMathSciNetGoogle Scholar
  14. Nishizawa, M. and Ueno, K. (2001). Integral solutions of hypergeometric q-difference systems with |q| = 1. In Physics and Combinatorics 1999 (Nagoya), pages 273–286. World Sci. Publishing, River Edge, NJ.Google Scholar
  15. Noumi, M. and Mimachi, K. (1992). Askey-Wilson polynomials as spherical functions on SUq(2). In Quantum Groups (Leningrad, 1990), volume 1510 of Lecture Notes in Math., pages 98–103. Springer, Berlin.Google Scholar
  16. Noumi, M. and Stokman, J. V. (2000). Askey-Wilson polynomials: an affine Hecke algebra approach. Preprint.Google Scholar
  17. Rosengren, H. (2000). A new quantum algebraic interpretation of the Askey-Wilson polynomials. In q-Series from a Contemporary Perspective (South Hadley, MA, 1998), volume 254 of Contemp. Math., pages 371–394. Amer. Math. Soc., Providence, RI.Google Scholar
  18. Ruijsenaars, S. N. M. (1997). First order analytic difference equations and integrable quantum systems. J. Math. Phys., 38(2):1069–1146.CrossRefzbMATHMathSciNetGoogle Scholar
  19. Ruijsenaars, S. N. M. (1999). A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type. Comm. Math. Phys., 206(3):639–690.zbMATHMathSciNetCrossRefGoogle Scholar
  20. Ruijsenaars, S. N. M. (2001). Special functions defined by analytic difference equations. In Bustoz, J., Ismail, M. E. H., and Suslov, S. K., editors, Special Functions 2000: Current Perspective and Future Directions, volume 30 of NATO Science Series, II. Math., Phys. and Chem., pages 281–333. Kluwer Acad. Publ.Google Scholar
  21. Stokman, J. V. (2001). Difference Fourier transforms for nonreduced root systems. Selecta Math. (N.S.). To appear. Available electronically: math.QA/0111221.Google Scholar
  22. Stokman, J. V. (2002). An expansion formula for the Askey-Wilson function. J. Approx. Theory, 114:308–342.CrossRefzbMATHMathSciNetGoogle Scholar
  23. Suslov, S. K. (1997). Some orthogonal very well poised 8ϕ7-functions. J. Phys. A, 30:5877–5885.CrossRefzbMATHMathSciNetGoogle Scholar
  24. Suslov, S. K. (2002). Some orthogonal very-well-poised 8ϕ7-functions that generalize Askey-Wilson polynomials. Ramanujan J., 5(2):183–218.CrossRefMathSciNetGoogle Scholar
  25. Van der Jeugt, J. and Jagannathan, R. (1998). Realizations of su(1, 1) and u q(su(1, 1)) and generating functions for orthogonal polynomials. J. Math. Phys., 39(9):5062–5078.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Jasper V. Stokman
    • 1
  1. 1.KdV Institute for MathematicsUniversiteit van AmsterdamAmsterdamThe Netherlands

Personalised recommendations