Lacunary formal power series and the Stern–Brocot sequence
Volume 159 / 2013
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 $\lambda_n = 2^{n+1} - 1$.
