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THE SECOND SHIFTED DIFFERENCE OF PARTITIONS AND ITS APPLICATIONS

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  25 August 2022

KEVIN GOMEZ Open the ORCID record for KEVIN GOMEZ [Opens in a new window] ,
JOSHUA MALES Open the ORCID record for JOSHUA MALES [Opens in a new window]  and
LARRY ROLEN Open the ORCID record for LARRY ROLEN [Opens in a new window]
KEVIN GOMEZ
Affiliation:
Department of Mathematics, Vanderbilt University, 1420 Stevenson Center, Nashville, TN 37240, USA e-mail: kevin.j.gomez@vanderbilt.edu
JOSHUA MALES*
Affiliation:
Department of Mathematics, University of Manitoba, 450 Machray Hall, Winnipeg, Canada
LARRY ROLEN
Affiliation:
Department of Mathematics, Vanderbilt University, 1420 Stevenson Center, Nashville, TN 37240, USA e-mail: larry.rolen@vanderbilt.edu
*
e-mail: joshua.males@umanitoba.ca
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Abstract

A number of recent papers have estimated ratios of the partition function $p(n-j)/p(n)$ , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions, $f(\,j,n) := p(n) -2p(n-j) +p(n-2j)$ , and give another easy-to-use estimate of $f(\,j,n)$ . As applications of these, we prove a shifted convexity property of $p(n)$ , as well as giving new estimates of the k-rank partition function $N_k(m,n)$ and non-k-ary partitions along with their differences.

Keywords

partitionconvexityasymptotic expansion

MSC classification

Primary: 11P82: Analytic theory of partitions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 1 , February 2023 , pp. 66 - 78
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The research of the second author conducted for this paper is supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute. This work was supported by a grant from the Simons Foundation (853830, LR). The third author is also grateful for support from a 2021–2023 Dean’s Faculty Fellowship from Vanderbilt University and to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.

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