Automatic sequences defined by Theta functions and some infinite products

Abstract

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 $$f(x)=\prod _{s=0}^{\infty }f(x^{q^s})=\sum _{i=0}^{\infty }{c}_i{x}^i,$$ and study when the sequence $(c_i{)}_{i\in \mathbf{N}}$ is $q$-automatic. The main result is that for given integers $q\ge 2$ and $d\ge 0$, there exist finitely many polynomials of degree $d$ defined over the field of rational numbers $\mathbf{Q}$, such that ${\prod }_{s=0}^{\infty }f(x^{q^s})={\sum }_{i=1}^{\infty }{c}_i{x}^i$ is a $q$-automatic power series.