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

Abstract

A result on the Galois theory of q-difference equations [3] leads to the following question: if q ∈ C * , |q| < 1 and if one sets θ q (z) := ∑ m∈Z q m(m−1)/2 z m , can some coefficients of the Laurent series expansion of θ k q (z), k ∈ N * , vanish ? We give a partial answer.