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

Volume 194 / 2020

Amanda Folsom, Min-Joo Jang, Sam Kimport, Holly Swisher Acta Arithmetica 194 (2020), 393-421 MSC: Primary 11P82, 11F37. DOI: 10.4064/aa190326-23-10 Published online: 3 April 2020


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 $R_n$.


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

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