Sur le développement en fraction continue d'une généralisation de la cubique de Baum et Sweet

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In 1976, Baum and Sweet gave the first example of a power series that is algebraic over the field F₂(T) and whose continued fraction expansion has partial quotients with bounded degree. This power series is the unique solution of the equation TX³+X-T=0. In 1986, Mills and Robbins described an algorithm that allows to compute the continued fraction expansion of the Baum--Sweet power series. In this paper, we consider the more general equations TXʳ⁺¹+X-T=0, where r is a power of a prime number p. Such an equation has a unique solution in the field Fₚ((T⁻¹)). Applying an approach already used by Lasjaunias, we give a description of the continued fraction expansion of these algebraic power series.