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