A Note on the Relation between Recognisable Series and Regular Sequences, and their Minimal Linear Representations

In this note, we precisely elaborate the connection between recognisable series (in the sense of Berstel and Reutenauer) and $q$-regular sequences (in the sense of Allouche and Shallit) via their linear representations. In particular, we show that the minimisation algorithm for recognisable series can also be used to minimise linear representations of $q$-regular sequences.