Infinite products related to generalized Thue-Morse sequences

Abstract

Given an integer $q\ge 2$ and ${\theta }_1,\cdots ,{\theta }_{q-1}\in \{0,1\}$, let $({\theta }_n{)}_{n\ge 0}$ be the generalized Thue-Morse sequence, defined to be the unique fixed point of the morphism $$0\mapsto 0{\theta }_1\cdots {\theta }_{q-1}$$ $$1\mapsto 1{\overline{\theta }}_1\cdots {\overline{\theta }}_{q-1}$$ beginning with ${\theta }_0:=0$, where $\overline{0}:=1$ and $\overline{1}:=0$. For rational functions $R$, we study infinite products of the forms $$\prod _{n=1}^{\infty }(R(n){)}^{(-1{)}^{{\theta }_n}}\text{and}\prod _{n=1}^{\infty }(R(n){)}^{{\theta }_n}.$$ This generalizes relevant results given by Allouche, Riasat and Shallit in 2019 on infinite products related to the famous Thue-Morse sequence $(t_n{)}_{n\ge 0}$ of the forms $$\prod _{n=1}^{\infty }(R(n){)}^{(-1{)}^{t_n}}\text{and}\prod _{n=1}^{\infty }(R(n){)}^{t_n}.$$