A heuristic guide to evaluating triple-sums

TL;DR

This paper introduces a heuristic method linking Appell--Lerch functions to divergent partial theta functions, enabling the evaluation of complex triple-sums in terms of special functions.

Contribution

It provides a heuristic approach to evaluate triple-sums using Appell--Lerch functions, extending previous methods for double-sums.

Findings

Heuristic relates Appell--Lerch functions to divergent partial theta functions.

Examples demonstrate evaluation of triple-sums via the heuristic.

Potential for expressing triple-sums in terms of false theta functions.

Abstract

Using a heuristic that relates Appell--Lerch functions to divergent partial theta functions one can expand Hecke-type double-sums in terms of Appell--Lerch functions. We give examples where the heuristic can be used as a guide to evaluate analogous triple-sums in terms of Appell--Lerch functions or false theta functions.