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A Fibonacci-Lucas based statistical model and several other related modelsare studied. The canonical and grand canonical partition functions for thesemodels are developed.Partition structure such as the distribution of sizes asin a cluster distribution is explored.Ensemble averaging over all partitionsleads to a scale invariant power law behavior at a particular critical likepoint. The canonical ensemble of the Fibonacci-Lucas case involves theGegenbauer polynomial.The model has a hyperbolic power law behavior, a featurelinked to the golden mean ratio of two adjacent Fibonacci numbers and also theconnection of Lucas numbers to the golden mean. The relation to other power lawbehavior, such as Zipf and Pareto laws, is mentioned. For the cases considered,the grand canonical ensemble involves the Gauss hypergeometric functionF(a,b,c,z) with specific values for a,b,c. The general case has a variablepower law behavior with tau exponent equal to1+c-a-b. An application witha=1/2, b=1, c=3 and thus tau=5/2 very closely approximates Bose-Einsteincondensation. The zeta function zeta(3/2)=2.61 of the exact theory is replacedwith 8/3 and zeta(5/2)= 1.34 with 4/3. At the condensation point the number ofcycles of length falls as a scale invariant power law. The cycles, which arisefrom permutation symmetries associated with Bose-Einstein statistics, can beviewed as links in a complex network of connections between particles. Thisscale invariant power law parallels that seen in complex networks. The growthof the network is developed using recurrence properties of the model.Constraints imposed by the canonical ensemble and associates correlations leadto some number theoretic connections between Fibonacci and Lucas numbers as anincidental consequence of this approach.

Power law behavior associated with a Fibonacci Lucas model and generalized statistical models