On the vanishing of coefficients of the powers of a theta function

TL;DR

This paper investigates whether coefficients in the Laurent series expansion of powers of a theta function can vanish, motivated by Galois theory of q-difference equations, providing partial answers to this mathematical question.

Contribution

It offers new insights into the behavior of coefficients of theta function powers, connecting Galois theory of q-difference equations with theta function analysis.

Findings

Partial conditions for vanishing coefficients identified

Connections established between Galois theory and theta functions

Results contribute to understanding the structure of theta function powers

Abstract

A result on the Galois theory of $q$-difference equations \cite{JSTALPAEN} leads to the following question: if $q \in \Cs$, $\lmod q \rmod < 1$ and if one sets $\thq(z) := \sum\limits_{m \in \Z} q^{m(m-1)/2} z^m$, can some coefficients of the Laurent series expansion of $\theta_q^k(z)$, $k \in \N^*$, vanish ? We give a partial answer.