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Dyson’s rank, overpartitions, and universal mock theta functions
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 64 / Issue 3 / September 2021
- Published online by Cambridge University Press:
- 07 September 2020, pp. 687-696
- Print publication:
- September 2021
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- Article
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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.
