Study of invariance of nonextensive statistics under the uniform energy spectrum translation

The general formalisms of the $q$-dual statistics, Boltzmann-Gibbs and three versions Tsallis statistics known as Tsallis-1, Tsallis-2, Tsallis-3 have been considered in canonical ensemble. We rigorously proved that probability distribution Tsallis-1 is invariant under uniform energy spectrum translation at a fixed temperature. This invariance demonstrates formalism consistent with fundamentals equilibrium statistical mechanics. same results we obtained for distributions statistics. However, found Tsallis-2 expectation values which are not normalization condition probabilities, indeed overall shift expected.