  The exact analytical formulas 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 normalized and Tsallis
unnormalized (also known as Tsallis-1 and Tsallis-2) statistics have been
consistently derived. The final exact results were expressed in terms of the
series expansions in the integral representation. The zeroth term approximation
to both quantum and classical statistics of particles has been introduced. We
have revealed that the phenomenological classical Tsallis distribution (widely
used in high energy physics) is equal to the distribution of the Tsallis
unnormalized statistics in the zeroth term approximation, but the
phenomenological quantum Tsallis distributions (introduced by definition on the
basis of the generalized entropy of the ideal gas) do not correspond to the
distributions of the Tsallis statistics. We have found that in the ranges of
the entropic parameter relevant to the processes of high-energy physics ($q<1$
for Tsallis-1 and $q>1$ for Tsallis-2) the Tsallis statistics is divergent.
Therefore, to obtain physical results, we have regularized the Tsallis
statistics by introducing an upper cut-off in the series expansion. The exact
numerical results for the Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann
statistics of particles in the Tsallis normalized and unnormalized statistics
have been obtained. We observed that the exact results of the Tsallis
statistics strongly enhanced the production of high-$p_{T}$ hadrons in
comparison with the usual phenomenological Tsallis distribution function at the
same values of $q$. The $q$-duality of the Tsallis normalized and unnormalized
statistics for the massive particles was studied.
