Now showing 1 - 10 of 12
- PublicationQuadratic forms and four partition functions modulo 3Recently, 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.
- PublicationMixed Mock Modular Q-seriesMixed 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.
- PublicationAutomorphic properties of generating functions for generalized rank moments and Durfee symbolsWe 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.
- PublicationRank and crank moments for overpartitionsWe 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).
Scopus© Citations 46 258
- PublicationM_2-rank differences for partitions without repeated odd partsWe 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.
Scopus© Citations 37 232
- PublicationM_2-rank differences for overpartitionsThis 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.
Scopus© Citations 19 252
- PublicationReal quadratic double sumsIn 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.
Scopus© Citations 7 212
- PublicationRank differences for overpartitionsIn 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.
Scopus© Citations 34 329
- PublicationQ-hypergeometric double sums as mock theta functionsRecently, 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.
279Scopus© Citations 13