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

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 ${\theta }_q^k(z)$, $k\in {\mathbb{N}}^{\ast }$, vanish ? We give a partial answer.

Submitted for publication

References in corpus (1)

• Théorie analytique locale des équations aux $q$-différences de pentes arbitraires