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

TL;DR

This paper explores the relationship between automata and nonhomogeneous linear recurrence systems over semirings, utilizing the $z$-transform to analyze their solutions and series representations.

Contribution

It establishes a correspondence between automata over a single-letter alphabet and nonhomogeneous linear recurrence systems, providing solutions and series representations using the $z$-transform.

Findings

Automata recognize specific nonhomogeneous recurrence systems.

The $z$-transform yields series representations of solutions.

Simplification methods for the $z$-transform series are presented.

Abstract

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.