The dual Hahnq-polynomials in the lattice and theq-algebras and

Abstract

The dual q-Hahn polynomials in the non uniform lattice x(s) = s] q s + 1] q are obtained. The main data for these polynomials are calculated ( the square of the norm the coe cients of the three term recurrence relation, etc), as well as its representation as a q-hypergeometric series. The connection with the Clebsch-Gordan Coe cients of the Quantum Algebras SU q (2) and SU q (1; 1) is also given. x1 IntroductionIt is well known that the Lie Groups Representation Theory plays a very important role in the Quantum Theory and in the Special Function Theory. The group theory is an effective tool for the investigation of the properties of di erent special functions, moreover, it gives the possibility to unify various special functions systematically. In a very simple and clear way, on the basis of group representation theory concepts, the Special Function Theory was developed in the classical book of N.Ya.Vilenkin 1] and in the monography of N.Ya.Vilenkin and A.U.Klimyk 2], which have an encyclopedic character.In recent years, the development of the quantum inverse problem method 3] and the study of solutions of the Yang-Baxter equations 4] gave rise to the notion of quantum groups and algebras, which are, from the mathematical point of view, Hopf algebras 5]. They are of great importance for applications in quantum integrable systems, in quantum eld theory, and statistical physics (see 6] and references contained therein). They are attracting much attention in quantum physics, especially after the introduction of the q-deformed oscillator 7]-8]. Also they have been used for the description of the rotational and vibrational spectra of deformed nuclei 9]-11] and diatomic molecules 12]-14], etc. However to apply them it is necessary to have a well developed theory of their representations. In quantum physics, for instance, the knowledge of the Clebsch-Gordan coe cients (3j symbols ), Racah coe cients ( 6j symbols ) and 9j symbols 15] is crucial for applications because all 1 today 2 e-mail address: renato@dulcinea.uc3m.es, Fax (+341) 6249430.

Quantum groups and their applications in nuclear physics

“…Clebsch-Gordan and Racah coefficients for su q (1,1) can be found in 91,94,96,98,118,119 . Furthemore, a two-parameter deformed version of su q (1,1), labelled as su p,q (1,1), has been introduced 53,106,120,121,122 .…”

The multiple sum formulas for 12j coefficients of SU(2) and uq(2)

“…with p = 3, x = q ±(c+1) , c = p+1 i=1 α i − p j=1 β j , as defined (with a minor correction) byÁlvarez-Nodarse and Smirnov 35 , instead of the standard basic hypergeometric functions p+1 φ p (see Gasper and Rahman 36 ). Parameters c = −1 and x = 1 for the balanced basic hypergeometric series, which appear in expressions for q-6j coefficients.…”

“…For the quantum algebra u q (2), the expansion of the q-9j coefficients in terms of q-6j coefficients was generalized by Nomura [29][30][31][32] and Smirnov et al 33 and extended to q-3nj coefficients (particularly, of the first and the second kind) by Nomura,30,31 who discussed their role in frames of the Yang-Baxter Relations. The corresponding summation formula of the twisted q-factorial series [generalizing Dougall's summation formula and resembling (but not equivalent with) the very well-poised basic hypergeometric 5 φ 4 series, depending on 3 parameters] needed for our purpose was derived by Ališauskas 34 and the twisted very well-poised q-factorial series, resembling the basic hypergeometric 7 φ 6 series (depending on 5 parameters) appear in a new approach 35 to the Clebsch-Gordan coefficients of u q (2). In the u q (3) context, Ališauskas 34 also used the summation formula of the q-factorial series depending on 4 parameters which correspond to Dougall's summation formula of the very well-poised hypergeometric 5 F 4 (1) or basic hypergeometric 6 φ 5 series.…”

The triple sum formulas for 9j coefficients of SU(2) and uq(2)

“…as defined ͑with a minor correction͒ by Á lvarezNodarse and Smirnov, 35 instead of the standard basic hypergeometric functions pϩ1 p ͑see Gasper and Rahman 44 ͒. Parameters cϭϪ1 and xϭ1 for the balanced basic hypergeometric series, which appear in expressions for q-6 j coefficients.…”

“…For the quantum algebra u q (2), the expansion of the q-9 j coefficients in terms of q-6 j coefficients was generalized by Nomura [29][30][31][32] and Smirnov et al 33 The corresponding summation a͒ Electronic mail: sigal@itpa.lt formula of the twisted q-factorial series ͓generalizing Dougall's summation formula and resembling ͑but not equivalent with͒ the very well-poised basic hypergeometric 5 4 series, depending on 3 parameters͔ needed for our purpose was derived by Ališauskas, 34 when the twisted very well-poised q-factorial series, resembling the basic hypergeometric 7 6 series ͑depending on 5 parameters͒ appear in a new approach 35 to the Clebsch-Gordan coefficients of u q (2).…”