Infinite products related to generalized Thue-Morse sequences (2006.04187v1)

Abstract: 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 $$\prod{n=1}^\infty\Big(R(n)\Big)^{(-1)^{t_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{t_n}.$$