Now showing 1 - 10 of 35
  • 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
    Two-dimensional lattices with few distances
    (European Mathematical Society, 2006-06) ;
    We prove that of all two-dimensional lattices of covolume 1 the hexagonal lattice has asymptotically the fewest distances. An analogous result for dimensions 3 to 8 was proved in 1991 by Conway and Sloane. Moreover, we give a survey of some related literature, in particular progress on a conjecture from 1995 due to Schmutz Schaller.
  • 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
    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.
      234Scopus© Citations 37
  • Publication
    M_2-rank differences for overpartitions
    (Polskiej Akademi Nauk, Instytut Matematyczny, 2010) ;
    This is the third and final installment in our series of papers applying the method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences. The study of rank differences was initiated by Atkin and Swinnerton-Dyer in their proof of Dyson’s conjectures concerning Ramanujan’s congruences for the partition function. Since then, other types of rank differences for statistics associated to partitions have been investigated. In this paper, we prove explicit formulas for M2-rank differences for overpartitions. Additionally, we express a third order mock theta function in terms of rank differences.
      257Scopus© Citations 19
  • Publication
    Representations of integers by certain positive definite binary quadratic forms
    (Springer, 2007-12) ;
    We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x2+Ny2 for a squarefree integer N.
  • Publication
    On a conjecture of Kimoto and Wakayama
    (American Mathematical Society, 2016-07) ; ;
    We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apery-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p-adic analysis.
      352Scopus© Citations 11
  • Publication
    Supercongruences satisfied by coefficients of 2F1 hypergeometric series
    Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular functions. We prove these two congruences using combinatorial properties of the coefficients.
  • 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
    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.
      282Scopus© Citations 13