Infinite products related to generalized Thue–Morse sequences

Abstract

Given an integer $q\ge 2$ and $\theta _1,\ldots ,\theta _{q-1}\in \{0,1\}$, let $(\theta _n)_{n\ge 0}$ be the generalized Thue–Morse sequence, defined to be the unique fixed point of the morphism

$$\begin{aligned} 0\mapsto & {} 0\theta _1\cdots \theta _{q-1}\\ 1\mapsto & {} 1\overline{\theta }_1\cdots \overline{\theta }_{q-1} \end{aligned}$$

beginning with $\theta _0:=0$, where $\overline{0}:=1$ and $\overline{1}:=0$. For ad hoc rational functions R, we evaluate infinite products of the forms

$$\begin{aligned} \prod _{n=1}^\infty \Big (R(n)\Big )^{(-1)^{\theta _n}}\quad \text {and}\quad \prod _{n=1}^\infty \Big (R(n)\Big )^{\theta _n}.\end{aligned}$$

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

$$\begin{aligned} \prod _{n=1}^\infty \Big (R(n)\Big )^{(-1)^{t_n}}\quad \text {and}\quad \prod _{n=1}^\infty \Big (R(n)\Big )^{t_n}. \end{aligned}$$