  We outline a new approach to calculating the quantum mechanical propagator in
the presence of geometrically non-trivial Dirichlet boundary conditions based
upon a generalisation of an integral transform of the propagator studied in
previous work (the so-called ``hit function''), and a convergent sequence of
Pad\'e approximants. In this paper the generalised hit function is defined as a
many-point propagator and we describe its relation to the sum over trajectories
in the Feynman path integral. We then show how it can be used to calculate the
Feynman propagator. We calculate analytically all such hit functions in $D=1$
and $D=3$ dimensions, giving recursion relations between them in the same or
different dimensions and apply the results to the simple cases of propagation
in the presence of perfectly conducting planar and spherical plates. We use
these results to conjecture a general analytical formula for the propagator
when Dirichlet boundary conditions are present in a given geometry, also
explaining how it can be extended for application for more general,
non-localised potentials. Our work has resonance with previous results obtained
by Grosche in the study of path integrals in the presence of delta potentials.
We indicate the eventual application in a relativistic context to determining
Casimir energies using this technique.
