Year: 2025
Type: article
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 ( ∑ r , s ≥ 0 − ∑ r , s > 0 ) ( − 1 ) r + s x r y s q a ( r 2 ) + b r s + c ( s 2 ) , \begin{equation*} \bigg ( \sum _{r,s\ge 0}-\sum _{r,s>0}\bigg )(-1)^{r+s}x^ry^sq^{a\binom {r}{2}+brs+c\binom {s}{2}}, \end{equation*} where a a , b b , and c c are positive integers with negative discriminant D ≔ b 2 − a c > 0 D≔b^2-ac>0 , into sums of products of theta functions and false theta functions. Here a general formula is obtained 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
DOI: 10.1090/spmj/1850
Author: E. Mortenson
Institution: St Petersburg University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4409934740