A general formula for Hecke-type false theta functions

TL;DR

This paper derives a comprehensive formula for Hecke-type false theta functions, unifying previous decompositions and extending the understanding of their structure in relation to theta functions and Appell functions.

Contribution

The authors present a general formula for Hecke-type double-sums that encompasses and extends prior decompositions into theta and false theta functions.

Findings

Unified decomposition formula for Hecke-type false theta functions

Extension of previous results to a broader class of sums

Connection to Appell functions and theta functions

Abstract

In recent work where Matsusaka generalizes the relationship between Habiro-type series and false theta functions after Hikami, five families of Hecke-type double-sums of the form \begin{equation*} \left( \sum_{r,s\ge 0 }-\sum_{r,s<0}\right)(-1)^{r+s}x^ry^sq^{a\binom{r}{2}+brs+c\binom{s}{2}}, \end{equation*} where $b^2-ac<0$, are decomposed into sums of products of theta functions and false theta functions. Here we obtain a general formula for such double-sums in terms of theta functions and false theta functions, which subsumes the decompositions of Matsusaka. Our general formula is similar in structure to the case $b^2-ac>0$, where Mortenson and Zwegers obtain a decomposition in terms of Appell functions and theta functions.