Given an integer $q\ge2$ and $\theta_1,\cdots,\theta_{q-1}\in\{0,1\}$, let $(\theta_n)_{n\ge0}$ be the generalized Thue-Morse sequence, defined to be the unique fixed point of the morphism $$0\mapsto0\theta_1\cdots\theta_{q-1}$$ $$1\mapsto1\overline{\theta}_1\cdots\overline{\theta}_{q-1}$$ beginning with $\theta_0:=0$, where $\overline{0}:=1$ and $\overline{1}:=0$. For rational functions $R$, we study infinite products of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{\theta_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{\theta_n}.$$ 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 $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{t_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{t_n}.$$

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与广义 Thue-Morse 序列相关的无穷积

给定一个整数 $q\ge2$ 和 $\theta_1,\cdots,\theta_{q-1}\in\{0,1\}$，让 $(\theta_n)_{n\ge0}$ 是广义的Thue-Morse 序列，定义为态射的唯一不动点 $$0\mapsto0\theta_1\cdots\theta_{q-1}$$$1\mapsto1\overline{\theta}_1\cdots\overline{\theta }_{q-1}$$ 以 $\theta_0:=0$ 开头，其中 $\overline{0}:=1$ 和 $\overline{1}:=0$。对于有理函数 $R$，我们研究 $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{\theta_n}}\quad 形式的无穷积\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{\theta_n}.$$ 这概括了 Allouche、Riasat 和 Shallit 在 2019 年在无穷大上给出的相关结果与著名的 Thue-Morse 序列 $(t_n)_{n\ge0}$ 相关的产品，形式为 $$\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}.$$