Now showing 1 - 10 of 12
  • Publication
    Automorphic properties of generating functions for generalized rank moments and Durfee symbols
    We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta functions.
  • Publication
    Mixed Mock Modular Q-series
    (Indian Mathematical Society, 2013-12) ;
    Mixed mock modular forms are functions which lie in the tensor space of mock modular forms and modular forms. As q-hypergeometric series, mixed mock modular forms appear to be much more common than mock theta functions. In this survey we discuss some of the ways such series arise.
  • Publication
    Quadratic forms and four partition functions modulo 3
    (De Gruyter, 2011-02) ;
    Recently, Andrews, Hirschhorn and Sellers have proven congruences modulo 3 for four types of partitions using elementary series manipulations. In this paper, we generalize their congruences using arithmetic properties of certain quadratic forms.
  • Publication
    Q-hypergeometric double sums as mock theta functions
    (Mathematical Science Publishers, 2013) ;
    Recently, Bringmann and Kane established two new Bailey pairs and used them to relate certain q-hypergeometric series to real quadratic fields. We show how these pairs give rise to new mock theta functions in the form of q-hypergeometric double sums. We also prove an identity between one of these sums and two classical mock theta functions introduced by Gordon and McIntosh.
      237Scopus© Citations 11
  • Publication
    Rank differences for overpartitions
    (Oxford University Press, 2008-06-02) ;
    In 1954, Atkin and Swinnerton-Dyer proved Dyson's conjectures on the rank of a partition by establishing formulae for the generating functions for rank differences in arithmetic progressions. In this paper, we prove formulae for the generating functions for rank differences for overpartitions. These are in terms of modular functions and generalized Lambert series.
      280Scopus© Citations 33
  • Publication
    On two 10th order mock theta identities
    (Springer, 2015-02) ;
    We give short proofs of conjectural identities due to Gordon and McIntosh involving two 10th order mock theta functions.
      195Scopus© Citations 2
  • Publication
    Rank and crank moments for overpartitions
    We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for some combinatorial functions which may be expressed in terms of the second moments. Finally, we establish a congruence modulo 3 involving one such combinatorial function and the Hurwitz class number H(n).
      208Scopus© Citations 44
  • Publication
    M_2-rank differences for partitions without repeated odd parts
    (Mathematics Institute Bordeaux, University Bordeaux 1, 2009) ;
    We prove formulas for the generating functions for M_2-rank differences for partitions without repeated odd parts. These formulas are in terms of modular forms and generalized Lambert series.
      175Scopus© Citations 34
  • Publication
    Real quadratic double sums
    (Elsevier, 2015-08) ;
    In 1988, Andrews, Dyson and Hickerson initiated the study of q-hypergeometric series whose coefficients are dictated by the arithmetic in real quadratic fields. In this paper, we provide a dozen q-hypergeometric double sums which are generating functions for the number of ideals of a given norm in rings of integers of real quadratic fields and prove some related identities.
      176Scopus© Citations 5
  • Publication
    Mock theta double sums
    (Cambridge University Press, 2016-06-10) ;
    We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.
      107Scopus© Citations 11