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

Zhang, Helen W. J.

Canadian mathematical bulletin, Tome 64 (2021) no. 3, pp. 687-696

Résumé

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.

Détail

DOI : 10.4153/S0008439520000752

Mots-clés : Overpartition, Dyson’s rank, Appell-Lerch sum, universal mock theta functions