  The analytical expressions for the ultrarelativistic transverse momentum
distributions of the Tsallis and the Tsallis-$2$ statistics were obtained. We
found that the transverse momentum distribution of the Tsallis-factorized
statistics, which is now largely used to describe the experimental transverse
momentum spectra of hadrons measured in $pp$ collisions at LHC and RHIC
energies, in the ultrarelativistic case is not equivalent to the transverse
momentum distributions of the Tsallis and the Tsallis-$2$ statistics. However,
we revealed that this distribution exactly coincides with the transverse
momentum distribution of the Tsallis-$2$ statistics in the zeroth term
approximation and is transformed to the transverse momentum distribution of the
Tsallis statistics in the zeroth term approximation by changing the parameter
$q$ to $1/q_{c}$. We demonstrated analytically on the basis of the
ultrarelativistic ideal gas that the Tsallis-factorized statistics is not
equivalent to the Tsallis and the Tsallis-$2$ statistics. In the present paper
the Tsallis statistics corresponds to the standard expectation values.
