  The generating functions for density matrix elements of the Jaynes-Cummings
model with cavity damping are analysed in terms of their eigenmodes, which are
characterised by a specific temporal behaviour. These eigenmodes are shown to
be proportional to particular generalised hypergeometric functions. The
relative weights of these eigenmodes in the generating functions are determined
by the initial conditions of the model. These weights are found by deriving
orthogonality relations involving adjoint modes. In an example it is shown how
the time-dependent density matrix elements and the related factorial moments
can be extracted from the eigenmode decompositions of the generating functions.
