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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.

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