  Studying the nature and consequences of electron-phonon interaction in
manganites is an area of intense ongoing research. Here, in an attempt to model
charge and orbital ordering in manganites displaying C-type
antiferromaganetism, we study cooperative Jahn-Teller effect in two-band
one-dimensional chains in the regimes of both strong and weak electron-phonon
couplings. These chains exhibit orbital ferromagnetism with only $d_{z^2}$
orbitals being occupied. At strong coupling and in the antiadiabatic regime,
using a controlled analytic nonperturbative treatment that accounts for the
quantum nature of the phonons, we derive the effective polaronic Hamiltonians
for a single chain as well as for interacting identically-long chains. Due to
cooperative effects, these effective Hamiltonians manifest a dominant
next-nearest-neighbor hopping compared to the usual nearest-neighbor hopping
and a significantly enhanced nearest-neighbor repulsion. For densities up to
half filling, upon tuning electron-phonon coupling, interacting-chain
[single-chain] Jahn-Teller systems undergo quantum phase transition from a
charge disordered state to a conducting charge-density-wave state characterized
by a wavevector $\vec{k} = (\pi ,\pi )$ [$k = \pi $]. On the other hand, up to
half filling, a weak coupling analysis reveals a transition from a disordered
state to an insulating charge-density-wave state with a wavevector that depends
linearly on the density; the ordering is analyzed within a Peierls instability
framework involving the dynamic noninteracting susceptibility at nesting
wavevector and phonon frequency. Our analysis provides an opportunity to
identify the regime of electron-phonon coupling in manganites through
experimentally determining the charge-ordering wavevector.
