  A novel approach to accounting for the influence of initial system-bath
correlations on the dynamics of an open quantum system, based on the
conventional projection operator technique, is suggested. To avoid the
difficulties of treating the initial correlations, the conventional
Nakajima-Zwanzig inhomogeneous generalized master equations (GMEs) for a
system's reduced statistical operator and correlation function are exactly
converted into the homogeneous GMEs (HGMEs), which take into account the
initial correlations in the kernel governing the evolution of these HGMEs. In
the second order (Born) approximation in the system-bath interaction, the
obtained HGMEs are local in time and valid at all timescales. They are further
specialized for a realistic equilibrium Gibbs initial (at $t=t_0$) system+bath
state (for a system reduced statistical operator an external force at $t>t_0$
is applied) and then for a bath of oscillators (Boson field). As an example,
the evolution of a selected quantum oscillator (a localized mode) interacting
with a Boson field (Fano-like model) is considered at different timescales. It
is shown explicitly how the initial correlations influence the oscillator
evolution process. In particular, it is shown that the equilibrium system's
correlation function acquires at the large timescale the additional constant
phase factor conditioned by survived initial system-bath correlations.
