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

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