Mathematical structures derived from the q-product uniquely determined by tsallis entropy

For a unified description of power-law behaviors such as chaos, fractal and scale-free network, Tsallis entropy has been applied to the generalization of the traditional Boltzmann-Gibbs statistics as a fundamental information measure. Tsallis entropy S_q is an one-parameter generalization of Shannon entropy S₁ in the sense that lim_qrarr1 S_q = S_i. The generalized Boltzmann-Gibbs statistics by means of Tsallis entropy is nowadays called Tsallis statistics. The main approach in Tsallis statistics has been the maximum entropy principle, but there have been missing some fundamental mathematical formulae such as law of error, q-Stirling´s formula and q-multinomial coefficient. Recently, we have succeeded in proving law of error in Tsallis statistics using the q-product uniquely determined by Tsallis entropy. Along the same lines as the proof, we present q-Stirling´s formula, q-multinomial coefficient and a conjecture on the q-central limit theorem in Tsallis statistics