  The production of the hidden-charm pentaquarks $P_{c}$ via pion-induced
reaction on a proton target is investigated within an effective Lagrangian
approach. Three experimentally observed states, $P_c(4312)$, $P_c(4440)$, and
$P_c(4457)$, are considered in the calculation, and the Reggeized $t$-channel
meson exchange is considered as main background for the reaction $\pi
^{-}p\rightarrow J/\psi n$. The numerical results show that the experimental
data of the total cross section of the reaction $\pi^{-}p\rightarrow J/\psi n$
at $W\simeq 5$ GeV can be well explained by contribution of the Reggeized $t$
channel with reasonable cutoff. If the branching ratios $Br[P_{c}\rightarrow
J/\psi N]\simeq 3\%$ and $Br[P_{c}\rightarrow \pi N]\simeq 0.05\%$ are taken,
the average value of the cross section from the $P_{c}(4312)$ contribution is
about 1.2 nb/100 MeV, which is consistent with existing rude data at
near-threshold energies. The results indicate that the branching ratios of the
$P_{c}$ states to the $J/\psi N$ and $\pi N$ should be small. The shape of
differential cross sections shows that the Reggeized $t$-channel provides a
sharp increase at extreme forward angles, while the differential cross sections
from the $P_{c}$ states contributions are relatively flat. High-precision
experimental measurements on the reaction $\pi ^{-}p\rightarrow J/\psi n$ at
near-threshold energies are suggested to confirm the LHCb hidden-charm
pentaquarks as genuine states, and such experiments are also helpful to
understand the origin of these resonance structures.
