Here, [F.sub.i]([r.sub.1]), [F.sub.k]([r.sub.2]) denote radial functions of the appropriate electron, [j.sub.1], [j.sub.2], [[pi].sub.1], [[pi].sub.2] denote the single particle spin and parity of the electrons, respectively, J is the total spin obtained by using the appropriate Clebsch-Gordan coefficients [2,10] and M denotes the magnetic quantum number of the total angular momentum,
In each quark configuration, spin and spatial angular momentum are coupled to a total single particle j-value and the Clebsch-Gordan coefficients determine the portion of spin-up and spin-down of the quark.
A comparative analysis of the properties of these polynomials and [su.sub.q](2) and [su.sub.q](1, 1) Clebsch-Gordan coefficients shows that each relation for q-Hahn polynomials has the corresponding partner among the properties of q-CGC and vice versa.
Clebsch-Gordan coefficients, discrete orthogonal polynomials (q-discrete orthogonal polynomials), Nikiforov-Uvarov approach, quantum groups and algebras
Based on this well known fact we investigate the relation between the Clebsch-Gordan coefficients -also known as 3 j symbols for the quantum algebras S[U.sub.q] (2) and S[U.sub.q] (1 ,1) with q-analogues of the Hahn polynomials on the non-uniform lattice x(s) = [q.sup.s] - 1 / q - 1.
Notice that (4.24) constitutes an extension--in this case a q-analog--of the well known relation between the classical Hahn polynomials and Clebsch-Gordan coefficients. Furthermore, this expression is in accordance with the results obtained in  (see also  for more details).
Now, the substitution of the parameters (4.22) into the expression (3.11) with the help of (4.24) gives the q-analog of the Racah formula for [su.sub.q] (2) Clebsch-Gordan coefficients Two particular cases are immediately derived from (4.25):
Relation between the Clebsch-Gordan coefficients for the quantum algebras [su.sub.q](2) and [su.sub.q](1, 1).