Rank deviations for overpartitions

TL;DR

This paper derives general formulas for overpartition rank deviations using Appell--Lerch series and theta functions, enabling the recovery of various existing overpartition rank difference identities.

Contribution

It introduces new general formulas for overpartition rank deviations expressed through special functions, facilitating the derivation of known identities.

Findings

Formulas in terms of Appell--Lerch series and theta functions

Ability to recover existing overpartition rank difference identities

Provides two illustrative examples

Abstract

We prove general fomulas for the deviations of two overpartition ranks from the average. 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 two illustrations.