Conclusions
We have shown that the Askey-Wilson polynomials of general form are generated by the algebra AW(3), which has a fairly simple structure and is the q-analog of a Lie algebra with three generators. The main properties of these polynomials (weight function, recursion relation, etc.) can be obtained directly from analysis of the representations of the algebra.
In this paper, we have considered finite-dimensional representations of the algebra AW(3) and the Aksey-Wilson polynomials of discrete argument corresponding to these representations. A separate analysis is required for the infinite-dimensional representations, which generate polynomials of a continuous argument (these polynomials were investigated in detail in the review [2]). Also of interest is investigation of representations of the algebra AW(3) for complex values of the basic parameter ω and of the structure parameters.
In our view, the algebra AW(3) by itself warrants careful study on account of several remarkable properties (in the first place, the duality with respect to the operators K0, K1) not present in the currently very popular quantum algebras of the type SUq(2).
We assume that the algebra AW(3) is an algebra of dynamical or “hidden” symmetry in all problems in which exponential or hyperbolic spectra and the corresponding q-polynomials arise. We hope that in time the algebra AW(3) will come to play the same role in “q-problems” as Lie algebras play in exactly solvable problems of quantum mechanics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Literature Cited
R. Askey and J. Wilson, SIAM J. Math. Anal.,10, 1008 (1979).
R. Askey and J. Wilson, Mem. Am. Math. Soc.,54, 1 (1985).
N. Y. Vilenkin, Special Functions and the Theory of Group Representations, AMS Translations of Math. Monogr., Vol. 22, Providence, R. I. (1968).
A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable [in Russian], Nauka, Moscow (1985).
L. L. Vaksman and Ya. S. Soibel'man, Funktsional Analiz i Ego Prilozhen.,22, 2 (1988).
T. H. Koornwinder, Preprint AM-R9013, Amsterdam (1990).
N. M. Atakishiev and S. K. Suslov, Teor. Mat. Fiz.,85, 64 (1990).
Ya. I. Granovskii and A. S. Zhedanov, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 5, 60 (1986).
A. S. Zhedanov, Teor. Mat. Fiz.,82, 11 (1990).
P. Feinsilver, Acta Appl. Math.,13, 291 (1988).
Ya. I. Granovskii and A. S. Zhedanov, Preprint 89-7 [in Russian], Physicotechnical Institute, Donetsk (1989).
E. D. Kagramanov et al., Preprint IC/89/42, Trieste (1989).
D. B. Fairlie, J. Phys. A,23, L183, (1990).
A. V. Odesskii, Funktsional Analiz i Ego Prilozhen.,20, 78 (1985).
E. K. Sklyanin, Funktsional Analiz i Ego Prilozhen.,16, 27 (1982);17, 34 (1983).
Ya. I. Granovskii and A. S. Zhedanov, Zh. Eksp. Teor. Fiz.,94 49 (1988).
Ya. I. Granovskii, A. S. Zhedanov, and I. M. Lutsenko, Zh. Eksp. Teor. Fiz.,99, 369 (1991).
Ya. I. Granovskii, A. S. Zhedanov, and I. M. Lutzenko, J. Phys. A. (1991) (in print).
L. C. Biedenharn, J. Phys. A,22, L873 (1989).
Additional information
Donetsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 89, No. 2, pp. 190–204, November, 1991
Rights and permissions
About this article
Cite this article
Zhedanov, A.S. “Hidden symmetry” of Askey-Wilson polynomials. Theor Math Phys 89, 1146–1157 (1991). https://doi.org/10.1007/BF01015906
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01015906