A general formula for Hecke-type false 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.

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References in corpus (5)

• A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
• The mixed mock modularity of certain duals of generalized quantum modular forms of Hikami and Lovejoy
• Hecke-Rogers double-sums and false theta functions
• A heuristic guide to evaluating triple-sums
• Witten-Reshetikhin-Turaev invariants and indefinite false theta functions for plumbing indefinite H-graphs