PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Lacunary formal power series and the Stern–Brocot sequence

Volume 159 / 2013

Jean-Paul Allouche, Michel Mendès France Acta Arithmetica 159 (2013), 47-61 MSC: Primary 11B83; Secondary 11B85, 11B65, 13F25, 11A55. DOI: 10.4064/aa159-1-3

Abstract

Let $F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-\lambda_n}$ be a real lacunary formal power series, where $\varepsilon_n = 0, 1$ and $\lambda_{n+1}/\lambda_n > 2$. It is known that the denominators $Q_n(X)$ of the convergents of its continued fraction expansion are polynomials with coefficients $0, \pm 1$, and that the number of nonzero terms in $Q_n(X)$ is the $n$th term of the Stern–Brocot sequence. We show that replacing the index $n$ by any $2$-adic integer $\omega$ makes sense. We prove that $Q_{\omega}(X)$ is a polynomial if and only if $\omega \in {\mathbb Z}$. In all the other cases $Q_{\omega}(X)$ is an infinite formal power series; we discuss its algebraic properties in the special case $\lambda_n = 2^{n+1} - 1$.

Authors

  • Jean-Paul AlloucheÉquipe Combinatoire et Optimisation
    CNRS, Institut de Mathématiques de Jussieu
    Université Pierre et Marie Curie
    Case 247, 4 Place Jussieu
    F-75252 Paris Cedex 05, France
    e-mail
  • Michel Mendès FranceMathématiques
    Université Bordeaux I
    F-33405 Talence Cedex, France
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image