  A nonhomogeneous system of linear recurrence equations can be recognized by
an automaton $\mathcal{A}$ over a one-letter alphabet $A = \{z\}$. Conversely,
the automaton $\mathcal{A}$ generates precisely this nonhomogeneous system of
linear recurrence equations. We present the solutions of these systems and
apply the $z$-transform to these solutions to obtain their series
representation. Finally, we show some results that simplify the series
representation of the $z$-transform of these solutions. We consider single
systems as well as the composition of two systems.
