A Final Coalgebra for k-regular Sequences

We study k-regular sequences from a coalgebraic perspective. Building on the observation that the set of streams over a semiring S can be turned into a final coalgebra, we obtain characterizations of kregular sequences in terms of finite weighted automata, finite systems of behavioral differential equations, and recognizable power series. The latter characterization is obtained via an isomorphism of final coalgebras based on the k-adic numeration system.