  We present a theoretical investigation for the full counting statistics of
the Andreev tunneling through a quantum dot (QD) embedded between
superconducting (SC) and normal leads in the presence of a strong on-site
electron-phonon interaction using nonequilibrium Green function method. For
this purpose, we generalize the dressed tunneling approximation (DTA) recently
developed in dealing with inelastic tunneling in a normal QD system to the
Andreev transport issue, which takes account of vibrational effect in
evaluation of electronic tunneling self energy in comparison with other simple
approaches and meanwhile allows us to derive an explicit analytical formula for
the cumulant generating function at the subgap region. We then analyze the
interplay of polaronic and SC proximity effects on the Andreev reflection
spectrum, current-voltage characteristics, and current fluctuations of the
hybrid system. Our main findings include: (1) no phonon side peaks in the
linear Andreev conductance; (2) a negative differential conductance stemming
from the suppressed Andreev reflection spectrum; (3) a novel inelastic resonant
peak in the differential conductance due to phonon assisted Andreev reflection;
(4) enhancement or suppression of shot noise for the symmetric or asymmetric
tunnel-coupling system respectively.
