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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Transformation properties for Dyson's rank function


Author: F. G. Garvan
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F37, 11P82; Secondary 05A19, 11B65, 11F11, 11P83, 11P84, 33D15
DOI: https://doi.org/10.1090/tran/7219
Published electronically: May 30, 2018
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Abstract: At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $ R(\zeta ,q)$, where $ R(z,q)$ is the two-variable generating function of Dyson's rank function and $ \zeta $ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon, and McIntosh, and motivated by Dyson's question, Bringmann, Ono, and Rhoades studied transformation properties of $ R(\zeta ,q)$. In this paper we strengthen and extend the results of Bringmann, Rhoades, and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo $ 5$ from the Lost Notebook has an analogue for all primes greater than $ 3$. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod $ 7$.


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Additional Information

F. G. Garvan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
Email: fgarvan@ufl.edu

DOI: https://doi.org/10.1090/tran/7219
Keywords: Dyson's rank function, Maass forms, mock theta functions, partitions, Mordell integral
Received by editor(s): June 17, 2016
Received by editor(s) in revised form: February 7, 2017
Published electronically: May 30, 2018
Additional Notes: The author was supported in part by a grant from the Simons Foundation (#318714). A preliminary version of this paper was first given on May 10, 2015, at the International Conference on Orthogonal Polynomials and $q$-series at the University of Central Florida, Orlando, in honour of Mourad Ismail’s 70th birthday.
Article copyright: © Copyright 2018 American Mathematical Society

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