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 Title: Quantum algebras {SU}_q(2) and {SU}_q(1,1) associated with certain q-Hahn polynomials: a revisited approach Author(s): Arvesú, Jorge Publisher: Kent State University Issued date: 2006 Citation: Electronic Transactions on Numerical Analysis, 2006, vol. 24, p. 24-44 URI: http://hdl.handle.net/10016/6435 ISSN: 1068-9613 Description: 21 pages, no figures.-- MSC2000 codes: 33D80, 16W35, 33D45, 81R50.MR#: MR2219911 (2007a:33017)Zbl 1109.33026 Abstract: This contribution deals with the connection of q-Clebsch-Gordan coefficients $(q$-CGC) of the Wigner-Racah algebra for the quantum groups $SU_q(2)$ and $SU_q(1,1)$ with certain q-Hahn polynomials. A comparative analysis of the properties of these polynomials and $su_q (2)$ and $su_q(1,1)$ Clebsch-Gordan coefficient shows that each relation for q-Hahn polynomials has the corresponding partner among the properties of q-CGC and vice versa. Consequently, special emphasis is given to the calculations carried out in the linear space of polynomials, i.e., to the main characteristics and properties for the new q-Hahn polynomials obtained here by using the Nikiforov-Uvarov approach [{\it A. F. Nikiforov}, {\it S. K. Suslov} and {\it V. B. Uvarov}, Orthogonal Polynomials in Discrete Variables, Springer-Verlag, Berlin, 1991; {\it A. F. Nikiforov} and {\it V. B. Uvarov}, Classical orthogonal polynomials in a discrete variable on non-uniform lattices, Preprint Inst. Prikl. Mat. M. V. Keldysh Akad. Nauk SSSR (In Russian), 17, Moscow, 1983] on the non-uniform lattice $x(s)=\frac{q -1}{q-1}$. These characteristics and properties will be important to extend the q-Hahn polynomials to the multiple case [{\it J. Arvesú}, q-Discrete multiple orthogonal polynomials, in preparation]. On the other hand, the aforementioned lattice allows to recover the linear one $x(s)=s$ as a limiting case, which does not happen in other investigated cases [{\it C. Campigotto}, {\it Yu. F. Smirnov} and {\it S. G. Enikeev}, q-Analogue of the Kravchuk and Meixner orthogonal polynomials, J. Comput. Appl. Math., 57, 87-97 (1995)]; A. del Sol Mesa and Yu. F. Smirnov, Clebsch-Gordon and Racah coefficients for $U_q(1,1)$ quantum algebra, (Discrete series). In Scattering, Reactions, Transitions in Quantum Systems and Symmetry Methods, R. M. Asherova and Yu. F. Smirnov, eds., 1991], for example in $x(s)=q 2s}$. This fact suggests that the q-analogues presented here (both from the point of view of quantum group theory and special function theory) are 'good' ones since all characteristics and properties, and consequently, all matrix element relations will converge to the standard ones when q tends to 1. Sponsor: The author was supported by the 'Ramón y Cajal' grant of the Ministerio de Ciencia y Tecnología of Spain as well as for the grant BFM2003-06335-C03-02. Review: PeerReviewed Publisher version: http://etna.mcs.kent.edu/vol.24.2006/pp24-44.dir/pp24-44.html Keywords: Clebsch-Gordan coefficientsDiscrete orthogonal polynomialsq-discrete orthogonal polynomialsNikiforov-Uvarov approachQuantum groups and algebras Rights: © Kent State University Appears in Collections: DM - GAMA - Artículos de Revistas