A method for obtaining a simple and compact expression for the generating function of the Weyl invariants is developed. The approach starts from the generating function of the representation basis in Bargmann space and uses the properties of this space in connection with Gaunt's integral. From the generating function of the Weyl invariants it is possible to derive the 3-j symbols of SU(N). When applied to SU(2), the method merely leads to the well-known Schwinger generating function of the 3-j symbols. The generating function of the SU(3) representation basis is built, and the SU(3) representation matrix elements are calculated; the equivalence between a part of these matrix elements and the Beg and Ruegg harmonic functions is proved. A new parametrisation for SU(N) is proposed, the invariant measure of which is explicitly determined. As an illustration, the approach is applied to the determination of the generation function of the Weyl invariants in some particular representations of SU(3) and SU(N).