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 a α → -1 (i.e., q → 1) 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.