Abstract
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.
Supported by the Royal Netherlands Academy of Arts and Sciences (KNAW).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Askey, R. and Wilson, J. A. (1985). Some basic hypergeometric polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 54(319).
Cherednik, I. (1997). Difference Macdonald-Mehta conjecture. Internat. Math. Res. Notices, 10:449–467.
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.
Gasper, G. and Rahman, M. (1990). Basic hypergeometric series, volume 35 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge.
Ismail, M. and Rahman, M. (1991). The associated Askey-Wilson polynomials. Trans. Amer. Math. Soc., 328:201–237.
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.
Koelink, E. (1996). Askey-Wilson polynomials and the quantum SU(2) group: survey and applications. Acta Appl. Math., 44(3):295–352.
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).
Koelink, E. and Stokman, J. V. (2001a). The Askey-Wilson function transform. Internat. Math. Res. Notices, 22:1203–1227.
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.
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.
Koornwinder, T. H. (1993). Askey-Wilson polynomials as zonal spherical functions on the su(2) quantum group. SIAM J. Math. Anal., 24:795–813.
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).
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.
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.
Noumi, M. and Stokman, J. V. (2000). Askey-Wilson polynomials: an affine Hecke algebra approach. Preprint.
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.
Ruijsenaars, S. N. M. (1997). First order analytic difference equations and integrable quantum systems. J. Math. Phys., 38(2):1069–1146.
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.
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.
Stokman, J. V. (2001). Difference Fourier transforms for nonreduced root systems. Selecta Math. (N.S.). To appear. Available electronically: math.QA/0111221.
Stokman, J. V. (2002). An expansion formula for the Askey-Wilson function. J. Approx. Theory, 114:308–342.
Suslov, S. K. (1997). Some orthogonal very well poised 8ϕ7-functions. J. Phys. A, 30:5877–5885.
Suslov, S. K. (2002). Some orthogonal very-well-poised 8ϕ7-functions that generalize Askey-Wilson polynomials. Ramanujan J., 5(2):183–218.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Stokman, J.V. (2005). Askey-Wilson Functions and Quantum Groups. In: Ismail, M.E., Koelink, E. (eds) Theory and Applications of Special Functions. Developments in Mathematics, vol 13. Springer, Boston, MA. https://doi.org/10.1007/0-387-24233-3_19
Download citation
DOI: https://doi.org/10.1007/0-387-24233-3_19
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-24231-6
Online ISBN: 978-0-387-24233-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)