Abstract
The oscillator quantum algebra is shown to provide a group-theoretic setting for the q-Laguerre and q-Hermite polynomials.
Similar content being viewed by others
References
Drinfel'd, V. G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Berkeley (1986), Vol. 1, Amer. Math. Soc., 1987, pp. 798–820.
JimboM., A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10, 63–69 (1985); A q-analogue of U(gl(N+1)), Hecke algebra and the Yang-Baxter equation, Lett. Math. Phys. 11, 247–252 (1986).
Woronowicz S. L., Compact matrix pseudogroups, Comm. Math. Phys. 111, 613–665 (1987).
Faddeev L. D., Reshetikhin N. Yu., and Takhatajan L. A., Quantization of Lie groups and Lie algebras, in Algebraic 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.
Gasper G. and Rahman M., Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
Exton H., q-Hypergeometric Functions and Applications, Ellis Horwood, Chichester, 1983.
Askey R., Ramanujan and hypergeometric and basic hypergeometric series, Russian Math. Surveys 45, 37–86 (1990).
Vilenkin N. Ya., Special Functions and the Theory of Group Representations, Amer. Math. Soc. Transl. of Math. Monographs 22, Amer. Math. Soc., Providence, 1968.
Miller W., Symmetry Groups and Their Applications, Academic Press, New York, 1972.
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 quantum groups and a q-analogue of orthogonal polynomials, C.R. Acad. Sci. Paris 307, 559–564 (1988).
Koornwinder T. H., Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials, Nederl. Akad. Wetensch. Proc. Ser. A 92, 97–117 (1989).
Vaksman L. L. and Soibelman Ya. S., An algebra of bounded functions on the quantum group of the motions of the plane, and q-analogues of Bessel functions, Soviet Math. Dokl. 39, 173–177 (1989).
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 of U 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, Lett. Math. Phys. 19, 195–204 (1990).
Kirillov A. N. and Reshetikhin N. Yu., Representations of the algebra U q (sl(2)), q-orthogonal polynomials and invariants of links, in V. G.Kac (ed.), Infinite Dimensional Lie Algebras and Groups, World Scientific, Singapore, 1989, pp. 285–339.
Noumi M. and Mimachi K., Quantum 2-spheres and big q-Jacobi polynomials, Comm. Math. Phys. 128, 521–531 (1990); Big q-Jacobi polynomials, q-Hahn polynomials, and a family of quantum 3-spheres, Lett. Math. Phys. 19, 299–305 (1990).
Biedenharn L. C., The quantum group SU(2) q and a q-analogue of the boson operators, J. Phys. A22, L873-L878 (1989).
Macfarlane A. J., On q-analogues of the quantum harmonic oscillator and the quantum group SU(2) q , J. Phys. A22, 4581–4588 (1989).
Sen C.-P. and Fu H.-C., The q-deformed boson realization of the quantum group SU(n) q and its representations, J. Phys. A22, L983-L986 (1989).
Floreanini R. and Vinet L., q-Analogues of the parabose and parafermi oscillators and representations of quantum algebras. J. Phys. A23, L1019-L1023 (1990).
Hayashi T., Q-analogue of Clifford and Weyl algebras-Spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127, 129–144 (1990).
Chaichan M. and Kulish P., Quantum Lie superalgebras and q-oscillators, Phys. Lett. B234, 72–80 (1990).
Floreanini R., Spiridonov V. P., and Vinet L., Bosonic realization of the quantum superalgebra osp q (1, 2n), Phys. Lett. B242, 383–386 (1990).
Floreanini R., Spiridonov, V. P., and Vinet, L., q-Oscillator realizations of the quantum superalgebras sl q (m, n) and osp q (m, 2n), Comm. Math. Phys., to appear.
D'Hoker, E., Floreanini, R., and Vinet, L., q-oscillator realizations of the metaplectic representation of quantum osp q (3, 2), J. Math. Phys., to appear.
Abe E., Hopf Algebras, Cambridge University Press, Cambridge, 1980.
Yan, H., q-Deformed oscillator algebra as a quantum group, Academia Sinica-preprint, AS-ITP-90-33 (1990).
Agarwal A. K., Kalnins E. G., and Miller W., Canonical equations and symmetry techniques for q-series, SIAM J. Math. Anal. 18, 1519–1538 (1987).
Moak D. S., The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81, 20–47 (1981).
Erdélyi A. (ed), Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.
Carlitz L., Some polynomials related to theta functions, Ann. Mat. Pura Appl. (4) 41, 359–373 (1955).
Carlitz L., Some polynomials related to theta functions, Duke Math. J. 24, 521–527 (1957).
Carlitz L., Note on orthogonal polynomials related to theta functions, Publicationes Mathematicae 5, 222–228 (1958).
Author information
Authors and Affiliations
Additional information
On leave from Laboratoire de Physique Nucléaire, Université de Montréal, Montréal, Canada H3C 3J7.