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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 powerexpansion of the infinite product$$f(x)=\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=0}^{\infty}c_ix^i,$$ and studywhen the sequence $(c_i)_{i \in \mathbf{N}}$ is $q$-automatic. The main resultis that for given integers $q \geq 2$ and $d \geq 0$, there exist finitely manypolynomials 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 $q$-automaticpower series.

Automatic sequences defined by Theta functions and some infinite products