General coefficient-vanishing results associated with theta series

TL;DR

This paper introduces two unified strategies to analyze coefficient-vanishing phenomena in theta series by examining products of theta functions under specific operators, providing both verification and discovery tools.

Contribution

It presents novel methods for studying coefficient-vanishing in theta series through the behavior of theta products under the huffing operator, unifying previous sporadic results.

Findings

Provides explicit criteria for the behavior of theta products

Enables verification of known coefficient-vanishing results

Facilitates discovery of new coefficient-vanishing phenomena

Abstract

There are a number of sporadic coefficient-vanishing results associated with theta series, which suggest certain underlying patterns. By expanding theta powers as linear combinations of products of theta functions, we present two strategies that will provide a unified treatment. Our approaches rely on studying the behavior of products of two theta series under the action of the huffing operator. For this purpose, some explicit criteria are given. We may use the presented methods to not only verify experimentally discovered coefficient-vanishing results, but also to produce a series of general phenomena.