  Let $p(x) \in C(x)$ be a rational function satisfying the condition $p(0)=1$
and $q$ an integer larger than $1$, in this article we will consider the power
expansion of the infinite product
$$f(x)=\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=0}^{\infty}c_ix^i,$$ and study
when the sequence $(c_i)_{i \in \mathbf{N}}$ is $q$-automatic. The main result
is that for given integers $q \geq 2$ and $d \geq 0$, there exist finitely many
polynomials of degree $d$ defined over the field of rational numbers
$\mathbf{Q}$, such that
$\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=1}^{\infty}c_ix^i$ is a $q$-automatic
power series.
