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

Jacques Sauloy; Changgui Zhang

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

Dynamical Systems 2020-08-03 v1 Classical Analysis and ODEs

Authors: Jacques Sauloy , Changgui Zhang

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.

Keywords

theta function q-difference equations galois theory laurent series vanishing coefficients

Cite

@article{arxiv.2007.16092,
  title  = {On the vanishing of coefficients of the powers of a theta function},
  author = {Jacques Sauloy and Changgui Zhang},
  journal= {arXiv preprint arXiv:2007.16092},
  year   = {2020}
}

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