  We construct a generating functional for the exact evalutation of a coherent
representation of spin network amplitudes. This generating functional is
defined for arbitrary graphs and depends only on a pair of spinors for each
edge. The generating functional is a meromorphic polynomial in the spinor
invariants which is determined by the cycle structure of the graph.
  The expansion of the spin network generating function is given in terms of a
newly recognized basis of SU(2) intertwiners consisting of the monomials of the
holomorphic spinor invariants. This basis is labelled by the degrees of the
monomials and is thus discrete. It is also overcomplete, but contains the
precise amount of data to specify points in the classical space of closed
polyhedra, and is in this sense coherent. We call this new basis the
discrete-coherent basis.
  We focus our study on the 4-valent basis, which is the first non-trivial
dimension, and is also the case of interest for Quantum Gravity. We find simple
relations between the new basis, the orthonormal basis, and the coherent basis.
  Finally we discuss the process of coarse graining moves at the level of the
generating functionals and give a general prescription for arbitrary graphs. A
direct relation between the polynomial of cycles in the spin network generating
functional and the high temperature loop expansion of the 2d Ising model is
found.
