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A general formula for Hecke-type false theta functions

Eric T. Mortenson

TL;DR

The paper addresses expressing Hecke-type double-sums $f_{a,b,c}(x,y;q)$ in the regime $D=b^2-ac<0$ as combinations of theta and false theta functions, extending Matsusaka's decompositions. It develops a general decomposition (Theorem mainTheorem) by leveraging elliptic transformation properties and the Jacobi triple product, and then validates the framework with explicit Habiro-type examples such as $H_p^{(2)}(q)$, including concrete cases like $H_{1}^{(2)}(q)$ and $H_{2}^{(2)}(q)$. The results provide a unified approach that mirrors the Appell-theta decompositions known for $D>0$ while yielding convergent partial-theta structures for $D<0$. This advances understanding of the modular-like structure and radial limits of Habiro-type series by linking double-sums to theta and false theta building blocks.

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.

A general formula for Hecke-type false theta functions

TL;DR

The paper addresses expressing Hecke-type double-sums in the regime as combinations of theta and false theta functions, extending Matsusaka's decompositions. It develops a general decomposition (Theorem mainTheorem) by leveraging elliptic transformation properties and the Jacobi triple product, and then validates the framework with explicit Habiro-type examples such as , including concrete cases like and . The results provide a unified approach that mirrors the Appell-theta decompositions known for while yielding convergent partial-theta structures for . This advances understanding of the modular-like structure and radial limits of Habiro-type series by linking double-sums to theta and false theta building blocks.

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 , 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 , where Mortenson and Zwegers obtain a decomposition in terms of Appell functions and theta functions.
Paper Structure (3 sections, 2 theorems, 22 equations)

This paper contains 3 sections, 2 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Proof of Theorem \ref{['theorem:mainTheorem']}

Key Result

Theorem 1.3

MZ Let $a,b,$ and $c$ be positive integers with $D:=b^2-ac>0$. For generic $x$ and $y$, we have where Here $d:=d^*+\{c/2 \}$ and $e:=e^*+\{ a/2\}$, with $0\le \{\alpha \}<1$ denoting fractional part of $\alpha$.

Theorems & Definitions (5)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['theorem:mainTheorem']}