Expressing $q$-series in terms of building blocks of Hecke-type double-sums

TL;DR

This paper rewrites recent double-sums and a $q$-hypergeometric function in terms of Hecke-type building blocks, analyzing their modularity properties and connecting them to mock modular forms.

Contribution

It introduces a new expression of double-sums and hypergeometric functions using Hecke-type blocks, clarifying their modularity and mock modularity aspects.

Findings

Expressed double-sums in terms of Hecke-type building blocks.

Determined (mock) modularity of the expressed sums.

Rewritten a $q$-hypergeometric function as a mixed mock modular form.

Abstract

We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent $q$-hypergeometric function of Andrews as a mixed mock modular form.