On certain recurrent and automatic sequences in finite fields

In this work we consider the general question: for a given algebraic formal power series with coefficients in a finite field, what kind of regularity (if any) can be expected for the partial quotients of the above power series in continued fraction expansion? Such a question is natural, since by a theorem of Christol, the coefficients of an algebraic power series over a finite field form an automatic sequence. Certain algebraic continued fractions are such that the sequence of the leading coefficients of the partial quotients is automatic. Here we give a rather general family of such sequences. Moreover, inspired by these examples, we give two criteria on automatic sequences, which allow us to obtain two new families of automatic sequences in an arbitrary finite field.