  In this paper we develop the Gerber-Shiu theory for the classic and dual
discrete risk processes in a Markovian (regime switching) environment. In
particular, by expressing the Gerber-Shiu function in terms of potential
measures of an upward (downward) skip-free discrete-time and discrete-space
Markov Additive Process (MAP), we derive closed form expressions for the
Gerber-Shiu function in terms of the so-called (discrete) $\boldsymbol{W}_v$
and $\boldsymbol{Z}_v$ scale matrices, which were introduced in
arXiv:2008.06697. We show that the discrete scale matrices allow for a unified
approach for identifying the Gerber-Shiu function as well as the value function
of the associated constant dividend barrier problems.
