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

Fluctuation theory for one-sided L\'evy processes with a matrix-exponential time horizon