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Mock modular forms and singular combinatorial series

Volume 159 / 2013

Amanda Folsom, Susie Kimport Acta Arithmetica 159 (2013), 257-297 MSC: 11F37, 11P82, 05A17. DOI: 10.4064/aa159-3-4

Abstract

A celebrated result of Bringmann and Ono shows that the combinatorial rank generating function exhibits automorphic properties after being completed by the addition of a non-holomorphic integral. Since then, automorphic properties of various related combinatorial families have been studied. Here, extending work of Andrews and Bringmann, we study general infinite families of combinatorial $q$-series pertaining to $k$-marked Durfee symbols, in which we allow additional singularities. We show that these singular combinatorial families are essentially mixed mock and quasimock modular forms, and provide their explicit non-holomorphic completions. As a special case of our work, we consider $k=3$, and provide an asymptotic expansion for the associated partition rank statistic, solving a special case of an open problem of Andrews.

Authors

  • Amanda FolsomMathematics Department
    Yale University
    10 Hillhouse Avenue
    P.O. Box 208283
    New Haven, CT 06520-8283, U.S.A.
    e-mail
  • Susie KimportMathematics Department
    Yale University
    10 Hillhouse Avenue
    P.O. Box 208283
    New Haven, CT 06520-8283, U.S.A.
    e-mail

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