dc.contributor.author Lovejoy, Jeremy dc.contributor.author Osburn, Robert dc.date.accessioned 2023-07-12T12:10:07Z dc.date.available 2023-07-12T12:10:07Z dc.date.issued 2023-07-12 dc.identifier.uri http://publications.mfo.de/handle/mfo/4054 dc.description.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 $6$. en_US dc.description.sponsorship The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for their support as this work began during their stay from January 9-22, 2022 as part of the Research in Pairs program. The second author was partially funded by a SSHN 2022 grant from the Embassy of France in Ireland during his visit to the Université Paris Cité from December 4-17, 2022. Finally, the authors are grateful to the Max-Planck-Institut für Mathematik for their hospitality and support during their joint stay from May 1-31, 2023. dc.language.iso en_US en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2023-11 dc.subject Overpartitions en_US dc.subject Rank en_US dc.subject M2-Rank en_US dc.subject Appell-Lerch series en_US dc.title Rank Deviations for Overpartitions en_US dc.type Preprint en_US dc.rights.license Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. de dc.rights.license This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties. en dc.identifier.doi 10.14760/OWP-2023-11 local.scientificprogram OWRF 2022 en_US local.series.id OWP-2023-11 en_US local.subject.msc 11 en_US local.subject.msc 05 en_US dc.identifier.urn urn:nbn:de:101:1-2024032009224643417989 dc.identifier.ppn 1858138884
﻿