The exact analytical formulae for the transverse momentum distributions of the Bose-Einstein, Fermi-Dirac, and Maxwell-Boltzmann statistics of particles with nonzero mass in the framework of the Tsallis-1, Tsallis-2 and q -dual statistics have been derived. The zeroth term approximation to both quantum and classical statistics of particles has been introduced.

We have found that the well-known phenomenological Tsallis distribution is equivalent to the classical transverse momentum distribution of the Tsallis-2 and q -dual statistics in the zeroth term approximation. The Tsallis-2 statistics is inconsistent. Therefore, the phenomenological Tsallis distribution is consistent with the basis of the statistical mechanics only if it belongs to the q-dual statistics. The exact transverse momentum distribution of the Tsallis-1 statistics was applied to describe the experimental data for proton-proton collisions at high energies.

We have shown the discrepancies in derivation of the Tsallis-like distribution function from the Tsallis statistics given in Buyukkilic et al (1995 Phys. Lett. A 197, 209).