  The last couple of years has seen a remarkable number of new, explicit
examples of the Wiener-Hopf factorization for Levy processes where previously
there had been very few. We mention in particular the many cases of spectrally
negative Levy processes, hyper-exponential and generalized hyper-exponential
Levy processes, Lamperti-stable processes, Hypergeometric processes,
Beta-processes and Theta-processes. In this paper we introduce a new family of
Levy processes, which we call Meromorphic Levy processes, or just M-processes
for short, which overlaps with many of the aforementioned classes. A key
feature of the M-class is the identification of their Wiener-Hopf factors as
rational functions of infinite degree written in terms of poles and roots of
the Levy-Khintchin exponent, all of which appear on the imaginary axis of the
complex plane. The specific structure of the M-class Wiener-Hopf factorization
enables us to explicitly handle a comprehensive suite of fluctuation identities
that concern first passage problems for finite and infinite intervals for both
the process itself as well as the resulting process when it is reflected in its
infimum. Such identities are of fundamental interest given their repeated
occurrence in various fields of applied probability such as mathematical
finance, insurance risk theory and queuing theory.
