  Let $\mathcal{P}(\Sigma^*)$ be the semiring of languages, and consider its
subset $\mathcal{P}(\Sigma)$. In this paper we define the language recognized
by a weighted automaton over $\mathcal{P}(\Sigma)$ and a one-letter alphabet.
Similarly, we introduce the notion of language recognition by linear recurrence
equations with coefficients in $\mathcal{P}(\Sigma)$. As we will see, these two
definitions coincide. We prove that the languages recognized by linear
recurrence equations with coefficients in $\mathcal{P}(\Sigma)$ are precisely
the regular languages, thus providing an alternative way to present these
languages. A remarkable consequence of this kind of recognition is that it
induces a partition of the language into its cross-sections, where the $n$th
cross-section contains all the words of length $n$ in the language. Finally, we
show how to use linear recurrence equations to calculate the density function
of a regular language, which assigns to every $n$ the number of words of length
$n$ in the language. We also show how to count the number of successful paths
of a weighted automaton.

Formal Languages and Automata Theory;Discrete Mathematics;Combinatorics