The $z$-Transform and Automata-Recognizable Systems of Nonhomogeneous Linear Recurrence Equations over Semirings

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.