论文标题
Apwenian序列的标准
Criteria for apwenian sequences
论文作者
论文摘要
在1998年，Allouche，Peyrière，Wen和Wen表明，以$ \ { - 1,1 \} $超过$ \ { - 1,1 \} $满足$ h_n/2^{n-1} \ equiv 1〜（\ equiv 1〜（\ Mathrm {modrm {mod} 〜2）$ n $ n $ n \ geq for laste $ h_n/2^{n-1^{受此结果的启发，fu和han引入了\ emph {apwenian}序列$ \ { - 1,1 \} $，即，$ \ pm 1 $序列的hankel decencens的hankel decents满足$ h_n/2^{n-1} {n-1} {n-1} \ equiv 1〜（equiv 1〜（equiv 1〜（equiv 1〜）序列是apwenian。在本文中，我们获得了易于检查的Apwenian序列的标准，这使我们能够确定所有是恒定长度替换的固定点的APWENIAN序列。令$ f（z）$为这样的apwenian序列的生成功能。我们表明，对于所有整数$ b \ ge 2 $，带有$ f（1/b）\ neq 0 $，实际数字$ f（1/b）$是先验的，其非理性指数等于$ 2 $。 此外，我们还得出了一个零一个apwenian序列的标准，其hankel决定因素满足$ h_n \ equiv 1〜（\ mathrm {mod} 〜2）$ for $ n \ geq 1 $。我们发现，在所有恒定长度的替换的固定点中，唯一的零一个apwenian序列是周期倍序序列。还给出了通过投影替换给出的Apwenian序列的各种示例。此外，我们证明了$ \ { - 1,1 \} $或$ \ {0,1 \} $上的所有Sturmian序列不是Apwenian。我们猜测，固定在$ \ { - 1,1 \} $或$ \ {0,1 \} $上的非稳定长度的固定点不能为apwenian。
In 1998, Allouche, Peyrière, Wen and Wen showed that the Hankel determinant $H_n$ of the Thue-Morse sequence over $\{-1,1\}$ satisfies $H_n/2^{n-1}\equiv 1~(\mathrm{mod}~2)$ for all $n\geq 1$. Inspired by this result, Fu and Han introduced \emph{apwenian} sequences over $\{-1,1\}$, namely, $\pm 1$ sequences whose Hankel determinants satisfy $H_n/2^{n-1}\equiv 1~(\mathrm{mod}~2)$ for all $n\geq 1$, and proved with computer assistance that a few sequences are apwenian. In this paper, we obtain an easy to check criterion for apwenian sequences, which allows us to determine all apwenian sequences that are fixed points of substitutions of constant length. Let $f(z)$ be the generating functions of such apwenian sequences. We show that for all integer $b\ge 2$ with $f(1/b)\neq 0$, the real number $f(1/b)$ is transcendental and its irrationality exponent is equal to $2$. Besides, we also derive a criterion for zero-one apwenian sequences whose Hankel determinants satisfy $H_n\equiv 1~(\mathrm{mod}~2)$ for all $n\geq 1$. We find that the only zero-one apwenian sequence, among all fixed points of substitutions of constant length, is the period-doubling sequence. Various examples of apwenian sequences given by substitutions with projection are also given. Furthermore, we prove that all Sturmian sequences over $\{-1,1\}$ or $\{0,1\}$ are not apwenian. And we conjecture that fixed points of substitution of non-constant length over $\{-1,1\}$ or $\{0,1\}$ can not be apwenian.
