Finite q-differences and the discrete renormalization group

The finite q-difference operator lends itself naturally to describing systems with exact discrete dilatation symmetry, such as hierarchical lattices. We express the homogeneity relation obeyed by the free energy in terms of the q-difference operator and integrate it to obtain the exact expression for the free energy. The site free energy distribution is multifractal, due to the topological nonuniformity of the hierarchical lattice, The Tsallis generalized entropy function provides a prescription for restoring the extensivity of the free energy.