$α$-divergence derived as the generalized rate function in a power-law system

The generalized binomial distribution in Tsallis statistics (power-law system) is explicitly formulated from the precise $q$-Stirling's formula. The $α$-divergence (or $q$-divergence) is uniquely derived from the generalized binomial distribution in the sense that when $α\rightarrow-1$ (i.e., $q\rightarrow1$) it recovers KL divergence obtained from the standard binomial distribution. Based on these combinatorial considerations, it is shown that $α$-divergence (or $q$-divergence) is appeared as the generalized rate function in the large deviation estimate in Tsallis statistics.

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