Transformation properties for Dyson’s rank function

Garvan, F.

doi:10.1090/tran/7219

Transformation properties for Dyson’s rank function
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by F. G. Garvan PDF

Trans. Amer. Math. Soc. 371 (2019), 199-248 Request permission

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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