  We investigate the interaction of correlated electrons with acoustical
phonons using the extended Hubbard-Holstein model in which both, the
electron-phonon interaction and the on-site Coulomb repulsion are considered to
be strong. The Lang-Firsov canonical transformation allows to obtain mobile
polarons for which a new diagram technique and generalized Wick's theorem is
used. This allows to handle the Coulomb repulsion between the electrons emerged
into a sea of phonon fields (\textit{phonon clouds}). The physics of emission
and absorption of the collective phonon-field mode by the polarons is discussed
in detail. Moreover, we have investigated the different behavior of optical and
acoustical phonon clouds when propagating through the lattice. In the
strong-coupling limit of the electron-phonon interaction, and in the normal as
well as in the superconducting phase, chronological thermodynamical averages of
products of acoustical phonon-cloud operators can be expressed by one-cloud
operator averages. While the normal one-cloud propagator has the form of a
Lorentzian, the anomalous one is of Gaussian form and considerably smaller.
Therefore, the anomalous electron Green's functions can be considered to be
more important than corresponding polarons functions, i.e., pairing of
electrons without phonon-clouds is easier to achieve than pairing of polarons
with such clouds.
