  In a rather general setting of It\^o-L\'evy processes we study a class of
transforms (Fourier for example) of the state variable of a process which are
holomorphic in some disc around time zero in the complex plane. We show that
such transforms are related to a system of analytic vectors for the generator
of the process, and we state conditions which allow for holomorphic extension
of these transforms into a strip which contains the positive real axis. Based
on these extensions we develop a functional series expansion of these
transforms in terms of the constituents of the generator. As application, we
show that for multidimensional affine It\^o-L\'evy processes with state
dependent jump part the Fourier transform is holomorphic in a time strip under
some stationarity conditions, and give log-affine series representations for
the transform.
