Journal of Statistical Physics, Vol. 47, Nos. 3/4, 1987. p. 297-330.
In the first two papers in this series we considered an extension of the hard hexagon model to a solvable two-dimensional lattice gas with at most two particles per pair of adjacent sites, and we described the local densities in terms of elliptic theta functions. Here we present the mathematical theory behind our derivation of the local densities. Our work centers on q-analogs of trinomial coefficients.