  There is an abundance of useful fluctuation identities for one-sided L\'evy
processes observed up to an independent exponentially distributed time horizon.
We show that all the fundamental formulas generalize to time horizons having
matrix exponential distributions, and the structure is preserved. Essentially,
the positive killing rate is replaced by a matrix with eigenvalues in the right
half of the complex plane which, in particular, applies to the positive root of
the Laplace exponent and the scale function. Various fundamental properties of
thus obtained matrices and functions are established, resulting in an easy to
use toolkit. An important application concerns deterministic time horizons
which can be well approximated by concentrated matrix exponential
distributions. Numerical illustrations are also provided.
