JEDNOSTKA NAUKOWA KATEGORII A+

# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## Acta Arithmetica

Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

## Quantum modular forms and singular combinatorial series with repeated roots of unity

### Tom 194 / 2020

Acta Arithmetica 194 (2020), 393-421 MSC: Primary 11P82, 11F37. DOI: 10.4064/aa190326-23-10 Opublikowany online: 3 April 2020

#### Streszczenie

In 2007, G. E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,\ldots ,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary two-variable partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, $R_n$ has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper complex half-plane $\mathbb H$, in work of Bringmann, Folsom, Garvan, Kimport, Mahlburg, and Ono. Quantum modular forms, defined by Zagier in 2010, are similar to modular or mock modular forms but are defined on the rationals $\mathbb Q$ as opposed to $\mathbb H$, and exhibit modular transformations there up to suitably analytic error functions in $\mathbb R$; in general, they have been related to diverse areas including number theory, topology, and representation theory. Here, we establish quantum modular properties of $R_n$.

#### Autorzy

• Amanda FolsomDepartment of Mathematics and Statistics
Amherst College
Amherst, MA 01002, U.S.A.
e-mail
• Min-Joo JangDepartment of Mathematics
The University of Hong Kong
Room 318, Run Run Shaw Building
Pokfulam, Hong Kong
e-mail
• Sam KimportDepartment of Mathematics
Stanford University
450 Serra Mall, Building 380
Stanford, CA 94305-2125, U.S.A.
e-mail
• Holly SwisherDepartment of Mathematics
Oregon State University
Kidder Hall 368
Corvallis, OR 97331-4605, U.S.A.
e-mail

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek