论文标题
stieltjes继续与纸折叠序列和rudin-shapiro序列相关的分数
Stieltjes continued fractions related to the Paperfolding sequence and Rudin-Shapiro sequence
论文作者
论文摘要
我们研究了由纸张折叠序列和鲁丁 - 夏普罗序列给出的两个持续分数。通过明确描述Conventents $ p_n（x）/q_n（x）$ modulo $ 4 $的某些子序列，我们提供了这两个持续分数的正式功率系列扩展（Modulo $ 4 $），并证明它们是一致的Modulo $ 4 $，以$ 4 $在$ \ Mathbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb {z} $ {x} [x] [x] [x] [x] [x] [x] [x] [x] [x] $。因此，正式功率系列扩展的系数序列为$ 2 $ - 自动。写$ q_ {n}（x）= \ sum_ {i \ ge 0} a_ {n，i} x^{i} $。然后$（q_ {n}（x））_ {n \ ge 0} $定义了二维系数序列$（a_ {n，i}）_ {n，i \ ge 0} $。我们证明了系数序列$（a_ {n，i} \ mod 4）_ {n \ ge 0} $由两个$（q_ {n}（x））_ {n \ ge 0} $和$（p_ {n}（n} n}（x）（x））_ {n \ ge 0} $ $ 2 $ 2 $ aut-automains $（q_ {n}（x））_ {n \ ge 0} $ $ automain。此外，这二维系数序列的图片Modulo $ 4 $呈现一种自相似现象。
We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents $P_n(x)/Q_n(x)$ modulo $4$, we give the formal power series expansions (modulo $4$) of these two continued fractions and prove that they are congruent modulo $4$ to algebraic series in $\mathbb{Z}[[x]]$. Therefore, the coefficient sequences of the formal power series expansions are $2$-automatic. Write $Q_{n}(x)=\sum_{i\ge 0}a_{n,i}x^{i}$. Then $(Q_{n}(x))_{n\ge 0}$ defines a two-dimensional coefficient sequence $(a_{n,i})_{n,i\ge 0}$. We prove that the coefficient sequences $(a_{n,i}\mod 4)_{n\ge 0}$ introduced by both $(Q_{n}(x))_{n\ge 0}$ and $(P_{n}(x))_{n\ge 0}$ are $2$-automatic for all $i\ge 0$. Moreover, the pictures of these two dimensional coefficient sequences modulo $4$ present a kind of self-similar phenomenon.
