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Osburn, Robert
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Osburn, Robert
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Osburn, Robert
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Now showing 1 - 7 of 7
- 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 - 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 - 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.
245Scopus© Citations 48 - 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.
235Scopus© Citations 8 - PublicationCongruences for Traces of Singular ModuliWe extend a result of Ahlgren and Ono [1] on congruences for traces of singular moduli of level 1 to traces defined in terms of Hauptmodul associated to certain groups of genus 0 of higher levels.
219Scopus© Citations 6 - PublicationSupercongruences for Apéry-like numbersIt is known that the numbers which occur in Apery's proof of the irrationality of zeta (2) have many interesting congruence properties while the associated generating function satisfies a second order differential equation. We prove supercongruences for a generalization of numbers which arise in Beukers' and Zagier's study of integral solutions of Apery-like differential equations.
322Scopus© Citations 14 - 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).
267Scopus© Citations 46