## On the continued fraction expansion of some formal power series. (Sur le développement en fraction continue de certaines séries formelles.)(French)Zbl 0657.10035

Let $$\mathbb F$$ be a finite field of characteristic $$p>2$$. According to a result of W. H. Mills and D. P. Robbins [J. Number Theory 23, 388–404 (1986; Zbl 0591.10021)] there exists a formal power series $$\alpha$$ with coefficients in $$\mathbb F$$ such that $$\alpha =ax+(b+\beta^{-1})^{-1},$$ with $$\beta =(\alpha^ p+f_{p- 2})/f_{p-1}$$ and $$f_ k=\sum_{0\leq j\leq k/2}\binom{k-j}{j} x^{k-2j}.$$ It is known [loc. cit.] that $$\alpha$$ has degree at most $$p+1$$ over $$\mathbb F(X)$$ and the sequence of partial quotients in the continued fraction expansion of $$\alpha$$ takes finitely many values.
In this note the author proves that this sequence is $$p$$-automatic i.e., is a fixed point of a $$p$$-substitution with constant length. This gives the only example actually known towards the conjecture of Mendès France saying that any formal power series over a finite field which is algebraic and has partial quotients with finitely many values is itself $$p$$-automatic.

### MSC:

 11J70 Continued fractions and generalizations 11J61 Approximation in non-Archimedean valuations 11B85 Automata sequences

Zbl 0591.10021