Transformation properties for Dyson's rank function

TL;DR

This paper investigates the transformation properties of Dyson's rank generating function, extending previous work to provide new proofs of classical conjectures and establishing analogues for all primes greater than three.

Contribution

It strengthens and extends existing results on the transformation properties of Dyson's rank function, leading to new proofs and generalizations of rank identities.

Findings

Provided a new proof of Dyson's rank conjecture.

Established an analogue of Ramanujan's Dyson rank identity for all primes greater than 3.

Extended transformation properties of the rank generating function.

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 $7$.