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Algebra i Analiz, 2024, Volume 36, Issue 1, Pages 195–203 (Mi aa1905)  

Research Papers

A general formula for Hecke-type false theta functions

E. T. Mortenson

Department of Mathematics and Computer Science, St. Petersburg State University, St. Petersburg, 199034, Russia
References:
Abstract: In recent work where Matsusaka generalized the relationship between Habiro-type series and false theta functions after Hikami, he decomposed 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 $a,b,$ and $c$ are positive integers with negative discriminant $D:=b^2-ac<0$, 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 of $D>0$, where Mortenson and Zwegers obtain a decomposition in terms of Appell functions and theta functions.
Keywords: mock theta functions, false theta functions, theta functions.
Funding agency Grant number
Foundation for the Development of Theoretical Physics and Mathematics BASIS 20-7-1-25-1
This work was supported by the Theoretical Physics and Mathematics Advancement Foundation BASIS, agreement No. 20-7-1-25-1.
Received: 12.02.2023
Document Type: Article
Language: English
Citation: E. T. Mortenson, “A general formula for Hecke-type false theta functions”, Algebra i Analiz, 36:1 (2024), 195–203
Citation in format AMSBIB
\Bibitem{Mor24}
\by E.~T.~Mortenson
\paper A general formula for Hecke-type false theta functions
\jour Algebra i Analiz
\yr 2024
\vol 36
\issue 1
\pages 195--203
\mathnet{http://mi.mathnet.ru/aa1905}
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