缺项形式幂级数与Stern-Brocot序列
Lacunary formal power series and the Stern-Brocot sequence
摘要：
设$F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-λ_n}$是实缺项形式幂级数，其中$\varepsilon_n = 0, 1$和$λ_{n+1}/λ_n > 2$ .已知其连分式展开式的收敛式的分解子$Q_n(X)$是系数为$0, \pm 1$的多项式，且$Q_n(X)$中非零项的个数是Stern-Brocot序列的第$n$项.我们表明，取代索引$n$由任何2-adic整数$ω$是有意义的。我们证明了$Q_ω(X)$是多项式当且仅当$ω\in {\mathbb Z}$。在所有其他情况下$Q_ω(X)$是一个无限形式的幂级数，我们讨论的代数性质的特殊情况$λ_n = 2^{n+1} - 1$。
英文摘要：
Let $F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-λ_n}$ be a real lacunary formal power series, where $\varepsilon_n = 0, 1$ and $λ_{n+1}/λ_n > 2$. 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 $ω$ makes sense. We prove that $Q_ω(X)$ is a polynomial if and only if $ω\in {\mathbb Z}$. In all the other cases $Q_ω(X)$ is an infinite formal power series, the algebraic properties of which we discuss in the special case $λ_n = 2^{n+1} - 1$.
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