Lacunary formal power series and the Stern-Brocot sequence

Jean-Paul Allouche ; Michel Mendès France

Acta Arithmetica, Tome 161 (2013), p. 47-61 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $F(X) = ∑_{n ≥ 0} (-1)^{εₙ} X^{-λₙ}$ be a real lacunary formal power series, where εₙ = 0,1 and $λ_{n+1}/λₙ > 2$. It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that $Q_{ω}(X)$ is a polynomial if and only if ω ∈ ℤ. In all the other cases $Q_{ω}(X)$ is an infinite formal power series; we discuss its algebraic properties in the special case $λₙ = 2^{n+1} - 1$.

Détail

Publié le : 2013-01-01

Zbl 1293.11046

EUDML-ID : urn:eudml:doc:279381