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.