Abstract
Two interpretations ofq-special functions based on quantum groups and algebras have been presented in the literature. The connection between these approaches is explained using as an example the case whereU q (sl(2)) is the basic structure.
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References
Drinfel'd, V. G., Quantum groups, inProceedings of the International Congress of Mathematicians, Berkeley (1986), vol. 1, The American Mathematical Society, Providence, 1987, pp. 798–820.
Jimbo, M., Aq-difference analogue of U(g) and the Yang-Baxter equation,Lett. Math. Phys. 10, 63–69 (1985); Aq-analogue of U(gl(N + 1)), Hecke algebra and the Yang-Baxter equation,ibid,11, 247-252 (1986).
Woronowicz, S. L., Compact matrix pseudogroups,Comm. Math. Phys. 111, 613–665 (1987).
Faddeev, L. D., Reshetikhin, N. Yu., and Takhtajan, L. A., Quantization of Lie groups and Lie algebras, inAlgebraic Analysis, vol. 1, Academic Press, New York, 1988, p. 129.
Manin, Yu. I., Quantum groups and non-commutative geometry, Centre de Recherches Mathematiques, Montréal, 1988.
Floreanini, R. and Vinet, L.,q-Orthogonal polynomials and the oscillator quantum group,Lett. Math. Phys. 22, 45–54 (1991).
Floreanini, R. and Vinet, L., The metaplectic representation of su q (1, 1) and theq-Gegenbauer polynomials,J. Math. Phys., to appear.
Floreanini, R. and Vinet, L.,q-Conformal quantum mechanics andq-special functions,Phys. Lett. B, to appear.
Floreanini, R. and Vinet, L., Quantum algebras andq-special functions, University of Montreal preprint, UdeM-LPN-TH54, 1991.
Floreanini, R. and Vinet, L., Addition formulas forq-Bessel functions, University of Montreal preprint, UdeM-LPN-TH60, 1991.
Floreanini, R. and Vinet, L., Representations of quantum algebras andq-special functions, in V. Dobrev and W. Scherer (eds),Proceedings of the II International Wigner Symposium, Springer-Verlag, Berlin, 1992.
Floreanini, R. and Vinet, L., Generalizedq-Bessel functions, University of Montreal preprint, UdeM-LPN-TH87, 1992.
Kalnins, E. G., Manocha, H. L., and Miller, W., Models ofq-algebra representations: I. Tensor products of special unitary and oscillator algebras,J. Math. Phys., to appear.
Kalnins, E. G., Miller, W., and Mukherjee, S., Models ofq-algebra representations: The group of plane motions, University of Minnesota preprint, 1992.
Miller, W.,Lie Theory and Special Functions, Academic Press, New York, 1968.
Miller, W., Lie theory andq-difference equations,SIAM J. Math. Anal. 1, 171–188 (1970).
Agarwal, A. K., Kalnins, E. G., and Miller, W., Canonical equations and symmetry techniques forq-series,SIAM J. Math. Anal. 18, 1519–1538 (1987).
Vaksman, L. L. and Soibelman, Ya. S., Algebra of functions of the quantum group SU(2),Funct. Anal. Appl. 22, 1–14 (1988).
Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., and Ueno, K., Representations of the quantum group SU q (2) and the littleq-Jacobi polynomials,J. Funct. Anal. 99, 357–386 (1991).
Koornwinder, T. H., Representations of the twisted SU(2) quantum group and someq-hypergeometric orthogonal polynomials,Nederl. Akad. Wetensch. Proc. Ser. A92, 97–117 (1989).
Vaksman, L. L. and Korogodskii, L. I., An algebra of bounded functions on the quantum group of the motions of the plane, andq-analogues of Bessel functions,Soviet Math. Dokl. 39, 173–177 (1989).
Koornwinder, T. H., The addition formula for littleq-Legendre polynomials and the SU(2) quantum group,SIAM J. Math. Anal. 22, 295–301 (1991).
Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Saburi, Y., and Ueno, K., Unitary representations of the quantum group SU q (1, 1): structure of the dual space ofU q (sl(2)),Lett. Math. Phys. 19, 187–194 (1990); Unitary representations of the quantum group SU q (1, 1): II-matrix elements of unitary representations and the basic hypergeometric functions,ibid. 19, 195-204 (1990).
Koelnik, H. T., On quantum groups andq-special functions, University of Leiden thesis, 1991.
Swarttouw, R., The Hahn-Extonq-Bessel functions, University of Delft thesis, 1992.
Gasper, G. and Rahman, M.,Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
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Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.
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Floreanini, R., Vinet, L. On the quantum group and quantum algebra approach toq-special functions. Lett Math Phys 27, 179–190 (1993). https://doi.org/10.1007/BF00739576
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DOI: https://doi.org/10.1007/BF00739576