  The non-local nature of the polaron formation in t-t'-t"-J model is studied
in large lattices up to 64 sites by developing a new numerical method. We show
that the effect of longer-range hoppings t' and t" is a large anisotropy of the
electron-phonon interaction (EPI) leading to a completely different influence
of EPI on the nodal and antinodal points in agreement with the experiments.
Furthermore, nonlocal EPI preserves polaron's quantum motion, which destroys
the antiferromagnetic order effectively, even at strong coupling regime,
although the quasi-particle weight in angle-resolved-photoemission spectroscopy
is strongly suppressed.
