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Higher order log-concavity of the overpartition function and its consequences

Part of: Combinatorics Error analysis and interval analysis Multiplicative number theory

Published online by Cambridge University Press:  03 April 2023

Gargi Mukherjee ,
Helen W. J. Zhang  and
Ying Zhong
Gargi Mukherjee
Affiliation:
Institute for Algebra, Science park 2, Johannes Kepler University, Altenberger Straße 69, Linz A-4040, Austria (gargi.mukherjee@dk-compmath.jku.at)
Helen W. J. Zhang
Affiliation:
School of Mathematics, Hunan University, Changsha 410082, People’s Republic of China (helenzhang@hnu.edu.cn) Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics, Changsha 410082, People’s Republic of China (YingZhong@hnu.edu.cn)
Ying Zhong
Affiliation:
School of Mathematics, Hunan University, Changsha 410082, People’s Republic of China (helenzhang@hnu.edu.cn)
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Abstract

Let ${\overline{p}}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher-order log-concavity property of the overpartition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove log-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient ${\overline{p}}(n-1){\overline{p}}(n+1)/{\overline{p}}(n)^2$ up to a certain order. This enables us to additionally prove 2-log-concavity and higher Turán inequalities with a unified approach.

Keywords

overpartition functionRademacher-type seriesr-log-concavityhigher order Turán inequalitieslog-concavity

MSC classification

Primary: 05A20: Combinatorial inequalities 11N37: Asymptotic results on arithmetic functions 65G99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 1 , February 2023 , pp. 164 - 181
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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