论文标题
与广义thue-morse序列有关的无限产品
Infinite products related to generalized Thue-Morse sequences
论文作者
论文摘要
给定一个整数$ q \ ge2 $和$θ_1，\ cdots，θ_{q-1} \ in \ {0,1 \} $，让$（θ_n）_ {n \ ge0} $是广义的thue-morse序列，定义为独特的固定点$$ 0 \mapSto0θ_1\cdotsθ_{q-1} $$ $$ 1 \ mapsto1 \overlineθ_1\ cdots \ cdots \overlineθ_{q-1} $$以$θ_0：= 0 $开头，$ \ + edline {0}：= 1 $ and $和$ \ overline = 0 $ = 0 $ = 0 $ = 0。对于理性功能$ r $，我们研究表格的无限产品$ \ prod_ {n = 1}^\ infty \ big（r（n）\ big）^{（ - 1）^{θ_n}} \ quad \ text {and} \ quad \ quad \ prod_ {n = 1} Riasat和Shallit在2019年对与著名的Thue-Morse序列$（T_N）_ {n \ GE0} $有关的无限产品$$ \ prod_ {n = 1}^\ infty \ big（r（n）\ big）^{（ - 1）^{t_n}} \ quad \ text {and} \ quad \ quad \ prod_ {
Given an integer $q\ge2$ and $θ_1,\cdots,θ_{q-1}\in\{0,1\}$, let $(θ_n)_{n\ge0}$ be the generalized Thue-Morse sequence, defined to be the unique fixed point of the morphism $$0\mapsto0θ_1\cdotsθ_{q-1}$$ $$1\mapsto1\overlineθ_1\cdots\overlineθ_{q-1}$$ beginning with $θ_0:=0$, where $\overline{0}:=1$ and $\overline{1}:=0$. For rational functions $R$, we study infinite products of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{θ_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{θ_n}.$$ This generalizes relevant results given by Allouche, Riasat and Shallit in 2019 on infinite products related to the famous Thue-Morse sequence $(t_n)_{n\ge0}$ of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{t_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{t_n}.$$
