Rank Deviations for Overpartitions

Abstract

We prove general fomulas for the deviations of two overpartition ranks from the average, namely \begin{equation*} \overline{D}(a, M) := \sum_{n \geq 0} \Bigl( \overline{N}(a, M, n) - \frac{\overline{p}(n)}{M} \Bigr) q^n \end{equation*} and \begin{equation*} \overline{D}{2}(a,M) := \sum{n \geq 0} \Bigl( \overline{N}{2}(a, M, n) - \frac{\overline{p}(n)}{M} \Bigr) q^n \end{equation*} where $\overline{N}(a, M, n)$ denotes the number of overpartitions of $n$ with rank congruent to $a$ modulo $M$, $\overline{N}{2}(a, M, n)$ is the number of overpartitions of $n$ with $M_2$-rank congruent to $a$ modulo $M$ and $\overline{p}(n)$ is the number of overpartitions of $n$. These formulas are in terms of Appell-Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the numerous overpartition rank difference identities in the literature. We give examples for $M=3$ and $6$.