- Osburn, Robert

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# Osburn, Robert

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- 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.67 - 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.
293 - 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.
78 - PublicationGaussian hypergeometric series and supercongruencesLet p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of Fp points on algebraic varieties and to Fourier coefficients of modular forms. In this paper, we explicitly determine these functions modulo higher powers of p and discuss an application to supercongruences. This application uses two non-trivial generalized Harmonic sum identities discovered using the computer summation package Sigma. We illustrate the usage of Sigma in the discovery and proof of these two identities.
206ScopusÂ© Citations 46 - PublicationThe first positive rank and crank moments for overpartitionsIn 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.
215ScopusÂ© Citations 9 - PublicationCongruences via modular formsWe prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apery-like differential equations.
194ScopusÂ© Citations 8 - PublicationRogers-Ramanujan type identities for alternating knotsWe highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogersâ€“Ramanujan type identities for alternating knots as conjectured by Garoufalidis, LÃª and Zagier.
243ScopusÂ© Citations 13 - PublicationOn sums of three squaresLet r3(n) be the number of representations of a positive integer n as a sum of three squares of integers. We give two distinct proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of r3(n).
105 - 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.
78 - 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.
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