  We study the differential cross section of the single inclusive $e^+e^-$
annihilation to the hadrons via $\gamma$-production, in the different
${k_{t}}$-factorization frameworks. In order to take into account the
transverse momenta of the incoming partons, for the first time, we apply the
Kimber et al (KMR) method to calculate the unintegrated parton fragmentation
functions (UFFs) from the ordinary integrated one, i.e., the parton
fragmentation functions (FFs), which satisfy the similar DGLAP evolution
equations, such as those of parton distribution functions (PDFs). Also, by
utilizing the different angular ordering constraints the results corresponding
to the Martin et al (MRW) in the leading order (LO) and the
next-to-leading-order (NLO) are obtained. The LO sets of DSS library for the
input FFs is used. The numerical results are compared with the experimental
data in the different energies which are reported by the different
collaborations, such as TASSO, AMY, MARK II, CELLO, DELPHI, SLD, ALEPH and
Belle with the other QCD+fragmentation models such as Pythia 6.4 and 8.2 parton
showers. The behaviors of the normalized differential cross sections and the
multiplicity versus the "transverse momentum" ($p_\bot$) are discussed. The
final results demonstrate that the KMR and MRW UFFs give a good description of
data and there is not much significant difference between the above three
schemes. On the other hand, our results become closer to the data for the lower
values of $p_\bot$ and the higher values of center of mass energies.
