Abstract
An algebraic interpretation of the one-variable quantum \(q\)-Krawtchouk polynomials is provided in the framework of the Schwinger realization of \(\fancyscript{U}_{q}(sl_{2})\) involving two independent \(q\)-oscillators. The polynomials are shown to arise as matrix elements of unitary “\(q\)-rotation” operators expressed as \(q\)-exponentials in the \(\fancyscript{U}_{q}(sl_{2})\) generators. The properties of the polynomials (orthogonality relation, generating function, structure relations, recurrence relation, difference equation) are derived by exploiting the algebraic setting. The results are extended to another family of polynomials, the affine \(q\)-Krawtchouk polynomials, through a duality relation.
Similar content being viewed by others
References
Atakishiyev, M.N., Groza, V.A.: The quantum algebra \(U_q(su_2)\) and \(q\)-Krawtchouk families of polynomials. J. Phys. A 37, 2625 (2004)
Biedenharn, L.C.: The quantum group \(SU_{q}(2)\) and a \(q\)-analogue of the boson operators. J. Phys. A 226, 873–878 (1989)
Cohen-Tannoudji, C., Diu, B., Laloë, F.: Mécanique Quantique. Hermann, Paris (1973)
Delsarte, P.: Association schemes and t-designs in regular semilattices. J. Combin. Theory Ser. A 20, 230–243 (1976)
Delsarte, P.: Properties and applications of the recurrence \(F(i+1, k+1, n+1)=q^{k+1}F(i, k+1, n)-q^{k}F(i, k, n)\). SIAM J. Appl. Math. 31, 262–270 (1976)
Dunkl, C.: An addition theorem for some \(q\)-Hahn polynomials. Monatsh. Math 85, 5–37 (1977)
Floreanini, R., Vinet, L.: Automorphisms of the \(q\)-oscillator algebra and basic orthogonal polynomials. Phys. Lett. A 180, 393–401 (1993)
Floreanini, R., Vinet, L.: On the quantum group and quantum algebra approach to \(q\)-special functions. Lett. Math. Phys. 27, 179–190 (1993)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge, MA (2004)
Gasper, G., Rahman, M.: Some systems of multivariable orthogonal \(q\)-Racah polynomials. Ramanujan J. 13, 389–405 (2007)
Genest, V.X., Vinet, L., Zhedanov, A.: The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states. J. Phys. A 46, 505203 (2013)
Genest, V.X., Vinet, L., Zhedanov, A.: Interbasis expansions for the isotropic 3D harmonic oscillator and bivariate Krawtchouk polynomials. J. Phys. A 47, 025202 (2014)
Geronimo, J.S., Iliev, P.: Bispectrality of multivariable Racah–Wilson polynomials. Constr. Approx. 311, 417–457 (2010)
Gilmore, R.: Lie Groups, Lie Algebras and Some of Their Applications. Dover Publications, New York (2006)
Griffiths, R.C.: Orthogonal polynomials on the multinomial distribution. Aus. J. Stat. 13, 27–35 (1971)
Iliev, P.: Bispectral commuting difference operators for multivariable Askey–Wilson polynomials. Trans. Am. Math. Soc. 363, 1577–1598 (2011)
Kalnins, E.G., Miller, W., Mukherjee, S.: Models of \(q\)-algebra representations: matrix elements of the \(q\)-oscillator algebra. J. Math. Phys. 34, 5333–5356 (1993)
Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues. Springer, Berlin (2010)
Koelink, E.: \(q\)-Krawtchouk polynomials as spherical functions on the Hecke algebra of type \(B\). Trans. Am. Math. Soc. 352, 4789–4813 (2000)
Koornwinder, T.: Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials. Indag. Math. (Proc.) 92, 97–117 (1989)
Koornwinder, T.: Additions to the formula lists. In: Koekoek, Lesky and Swarttouw (eds.) Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogue. arXiv:1401.0815, (2014)
Miller, W., Kalnins, E.G., and Mukherjee, S.: Models of \(q\)-algebra representations: Matrix elements of \(U_{q}(su_2)\). In: Lie Algebras, Cohomology and New Applications to Quantum Mechanics, Contemporary Mathematics. American Mathematical Society (1994)
Smirnov, Y., Campigotto, C.: The quantum \(q\)-Krawtchouk and \(q\)-Meixner polynomials and their related \(D\)-functions for the quantum groups \(SU_q(2)\) and \(SU_q(1,1)\). J. Comp. Appl. Math. 164–165, 643–660 (2004)
Stanton, D.: A partially ordered set and q-Krawtchouk polynomials. J. Combin. Theory Ser. A 30, 276–284 (1981)
Stanton, D.: Orthogonal polynomials and Chevalley groups. In: Askey, R.A., Koornwinder, T.H., Schempp, W. (eds.) Special Functions: Group Theoretical Aspects and Applications. Mathematics and Its Applications, vol. 18, pp. 87–128. Springer, Netherlands (1984). doi:10.1007/978-94-010-9787-1_2
Tratnik, M.V.: Some multivariable orthogonal polynomials of the Askey tableau-discrete families. J. Math. Phys. 32, 2337 (1991)
Truax, D.R.: Baker-Campbell-Hausdorff relations and unitarity of \(SU(2)\) and \(SU(1,1)\) squeeze operators. Phys. Rev. D 31, 1988–1991 (1985)
Vilenkin, N.J., Klimyk, A.U.: Representation of Lie Groups and Special Functions. Kluwer Academic Publishers, Dordrecht (1991)
Zhedanov, A.: \(Q\) rotations and other \(Q\) transformations as unitary nonlinear automorphisms of quantum algebras. J. Math. Phys. 34, 2631 (1993)
Acknowledgments
L.V. wishes to acknowledge the hospitality of the Shanghai Jiao Tong University where this research project was initiated. V.X.G and L.V. would like to acknowledge the support provided to them by the University of Hawai’i, where this research was completed. V.X.G. holds an Alexander-Graham-Bell fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC). The research of L.V. is supported in part by NSERC.
Author information
Authors and Affiliations
Corresponding author
Additional information
V. X. Genest and L. Vinet were supported in part by NSERC.
Rights and permissions
About this article
Cite this article
Genest, V.X., Post, S., Vinet, L. et al. \(q\)-Rotations and Krawtchouk polynomials. Ramanujan J 40, 335–357 (2016). https://doi.org/10.1007/s11139-015-9681-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-015-9681-0