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Transactions of the American Mathematical Society

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The generating function of the $ M_2$-rank of partitions without repeated odd parts as a mock modular form


Author: Chris Jennings-Shaffer
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11P81, 11P82
DOI: https://doi.org/10.1090/tran/7212
Published electronically: May 9, 2018
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Abstract: By work of Bringmann, Ono, and Rhoades it is known that the generating function of the $ M_2$-rank of partitions without repeated odd parts is the so-called holomorphic part of a certain harmonic Maass form. Here we improve the standing of this function as a harmonic Maass form and show more can be done with this function. In particular, we show the related harmonic Maass form transforms like the generating function for partitions without repeated odd parts (which is a modular form). We then use these improvements to determine formulas for the rank differences modulo $ 7$. Additionally, we give identities and formulas that allow one to determine formulas for the rank differences modulo $ c$, for any $ c>2$.


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Additional Information

Chris Jennings-Shaffer
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: chrisjenningsshaffer@gmail.com

DOI: https://doi.org/10.1090/tran/7212
Received by editor(s): March 30, 2016
Received by editor(s) in revised form: December 13, 2016, and February 8, 2017
Published electronically: May 9, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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