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Dyson’s rank, overpartitions, and universal mock theta functions

Part of: Sequences and sets Discontinuous groups and automorphic forms Additive number theory; partitions

Published online by Cambridge University Press:  07 September 2020

Helen W. J. Zhang
Helen W. J. Zhang*
Affiliation:
School of Mathematics, Hunan University, Changsha410082, People's Republic of China
*
e-mail: [email protected]
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Abstract

In this paper, we decompose $\overline {D}(a,M)$ into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let $\overline {p}(n)$ be the number of partitions of n and $\overline {N}(a,M,n)$ be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average:

$$ \begin{align*} \overline{D}(a,M) &=\sum\limits_{n=0}^{\infty}\Big(\overline{N}(a,M,n) -\frac{\overline{p}(n)}{M}\Big)q^{n}. \end{align*} $$
Based on Appell–Lerch sum properties and universal mock theta functions, we obtain the stronger version of the results of Bringmann and Lovejoy.

Keywords

OverpartitionDyson’s rankAppell-Lerch sumuniversal mock theta functions

MSC classification

Primary: 11P84: Partition identities; identities of Rogers-Ramanujan type
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities 11F11: Holomorphic modular forms of integral weight 11F27: Theta series; Weil representation; theta correspondences
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 687 - 696
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author would like to thank the referees for their valuable comments and suggestions. This work was supported by the National Science Foundation of China (Grant Nos. 11871370 and 12001182) and the Fundamental Research Funds for the Central Universities (Grant No. 531118010411).

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