Lacunary formal power series and the Stern–Brocot sequence

Jean-Paul Allouche ¹   ; Michel Mendès France ²

¹ Équipe 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
² Mathématiques Université Bordeaux I F-33405 Talence Cedex, France

Acta Arithmetica, Tome 159 (2013) no. 1, pp. 47-61

Résumé

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 $\lambda_n = 2^{n+1} - 1$.

Détail

DOI : 10.4064/aa159-1-3

Keywords: sum geq varepsilon lambda real lacunary formal power series where varepsilon lambda lambda known denominators convergents its continued fraction expansion polynomials coefficients number nonzero terms nth term stern brocot sequence replacing index adic integer omega makes sense prove omega polynomial only omega mathbb other cases omega infinite formal power series discuss its algebraic properties special lambda