- Osburn, Robert

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# Osburn, Robert

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- 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.
382 - PublicationTwo-dimensional lattices with few distancesWe 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.
160 - 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.
290 - 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.
234ScopusÂ© Citations 37 - 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 Dysonâ€™s conjectures concerning Ramanujanâ€™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.
257ScopusÂ© Citations 19 - PublicationRepresentations of integers by certain positive definite binary quadratic formsWe 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.
219 - PublicationOn a conjecture of Kimoto and WakayamaWe 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.
352ScopusÂ© Citations 11 - PublicationSupercongruences satisfied by coefficients of 2F1 hypergeometric series(Association Mathematique du Quebec, 2010)
; ; ; 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.193 - 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.
101 - 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.
282ScopusÂ© Citations 13