  Based on a previously developed recursive approach for calculating the
short-time expansion of the propagator for systems with time-independent
potentials and its time-dependent generalization for simple single-particle
systems, in this paper we present a full extension of this formalism to a
general quantum system with many degrees of freedom in a time-dependent
potential. Furthermore, we also present a recursive approach for the
velocity-independent part of the effective potential, which is necessary for
calculating diagonal amplitudes and partition functions, as well as an
extension from the imaginary-time formalism to the real-time one, which enables
to study the dynamical properties of quantum systems. The recursive approach
developed here allows an analytic derivation of the short-time expansion to
orders that have not been accessible before, using the implemented SPEEDUP
symbolic calculation code. The analytically derived results are extensively
numerically verified by treating several models in both imaginary and real
time.
