Generalized rank deviations for overpartitions

TL;DR

This work addresses the problem of obtaining explicit formulas for generalized rank deviations $Dbar_d(a,M)$ for overpartitions. It develops an Appell-Lerch series framework, expressing the generating functions via $m(x,q,z)$ and theta-quotient data, with parity-based decompositions that unify and extend previous results. The authors prove two main theorems that reduce the deviations to combinations of Appell-Lerch terms and related functions, and they show how to deduce single-deviation formulas from pairwise identities. As an application, they compute a 3-dissection of $O_d(zeta_3;q)$, yielding an explicit modular-analytic decomposition that demonstrates the utility of their approach for dissecting rank-type generating functions in arithmetic progressions.

Abstract

We prove formulas for generalized rank deviations for overpartitions. These formulas are in terms of Appell-Lerch series and sums of quotients of theta functions and extend work of Lovejoy and the second author. As an application, we compute a dissection.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Proposition 2.4
• proof