Extensive Rényi statistics from non-extensive entropy

We show that starting with either the non-extensive Tsallis entropy in Wang's formalism or the extensive Renyi entropy, it is possible to construct equilibrium non-Gibbs canonical distribution functions which satisfy the fundamental equations of thermodynamics. The statistical mechanics with Tsallis entropy does not satisfy the zeroth law of thermodynamics at dynamical and statistical independence request, whereas the extensive Renyi statistics fulfills all requirements of equilibrium thermodynamics in the microcanonical ensemble. Transformation formulas between Tsallis statistics in Wang representation and Renyi statistics are presented. The one-particle distribution function in Renyi statistics for a classical ideal gas at finite particle number has a power-law tail for large momenta.