  In this and subsequent paper arXiv:1011.5185 we develop a recursive approach
for calculating the short-time expansion of the propagator for a general
quantum system in a time-dependent potential to orders that have not yet been
accessible before. To this end the propagator is expressed in terms of a
discretized effective potential, for which we derive and analytically solve a
set of efficient recursion relations. Such a discretized effective potential
can be used to substantially speed up numerical Monte Carlo simulations for
path integrals, or to set up various analytic approximation techniques to study
properties of quantum systems in time-dependent potentials. The analytically
derived results are numerically verified by treating several simple models.
