Normalized U-q(sl(3)) Gel'fand-(Weyl)-Zetlin basis and new summation formulas for q-hypergeometric functions

An explicit realization of the normalized Gel'fand-(Weyl)-Zetlin (GWZ) basi s for U-q(sl(3)) in terms of polynomial functions in three variables (real or complex) is given. The construction uses two different realizations of t he U-q(sl(3)) unnormalized GWZ basis which were given previously, and whose transformation properties were not known. It turns out that finding these properties enables us to find the (GWZ-dependent) proportionality constant between these two realizations. The scalar product is also fixed by this in both unnormalized realizations, and then, by normalization, the normalized GWZ states are obtained. As by-products new summation formulas are obtaine d which seem new also for q=1. The main new formula is a double sum which i s given in terms of the proportionality constant mentioned above. This doub le sum can be written as single sum over a q-F-3(2) hypergeometric function , or as a q-hypergeometric function of two variables. (C) 2000 American Ins titute of Physics. [S0022-2488(00)00511-9].