Revisiting regular sequences in light of rational base numeration systems

Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract numeration system. When the numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated by its $S$-kernel is finitely generated. In this paper, we give a new characterization of such sequences in terms of the underlying numeration tree $T(L)$ whose nodes are words of $L$. We may decorate these nodes by the sequence of interest following a breadth-first enumeration. For a prefix-closed regular language $L$, we prove that a sequence is $S$-regular if and only if the tree $T(L)$ decorated by the sequence is linear, i.e., the decoration of a node depends linearly on the decorations of a fixed number of ancestors. Next, we introduce and study regular sequences in a rational base numeration system, whose numeration language is known to be highly non-regular. We motivate and comment our definition that a sequence is $\frac{p}{q}$-regular if the underlying numeration tree decorated by the sequence is linear. We give the first few properties of such sequences, we provide a few examples of them, and we propose a method for guessing $\frac{p}{q}$-regularity. Then we discuss the relationship between $\frac{p}{q}$-automatic sequences and $\frac{p}{q}$-regular sequences. We finally present a graph directed linear representation of a $\frac{p}{q}$-regular sequence. Our study permits us to highlight the places where the regularity of the numeration language plays a predominant role.