# Automatic sequences defined by Theta functions and some infinite products

## TL;DR

This paper investigates conditions under which infinite products involving rational functions and theta functions produce $q$-automatic sequences, establishing finiteness results for such polynomials over the rationals.

## Contribution

It proves that for fixed q and degree d, only finitely many polynomials over Q generate $q$-automatic sequences from these infinite products.

## Key findings

- Finiteness of polynomials producing $q$-automatic sequences
- Characterization of sequences from infinite products involving rational functions
- Conditions for automaticity in power series derived from theta functions

## 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 $$f(x)=\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=0}^{\infty}c_ix^i,$$ 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.