Lacunary formal power series and the Stern-Brocot sequence

TL;DR

This paper explores the properties of lacunary formal power series related to the Stern-Brocot sequence, extending the analysis to 2-adic indices and examining the algebraic nature of associated polynomials and series.

Contribution

It introduces a novel extension of the denominators of continued fraction convergents to 2-adic indices and characterizes when these are polynomials versus infinite series.

Findings

Q_ω(X) is a polynomial iff ω is an integer

For non-integer ω, Q_ω(X) is an infinite formal power series

Special case analysis when λ_n = 2^{n+1} - 1

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, the algebraic properties of which we discuss in the special case $\lambda_n = 2^{n+1} - 1$.