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Inequality arising from the iterated Laguerre operator for various partitions

Part of: Additive number theory; partitions Zeta and $L$-functions: analytic theory Combinatorics

Published online by Cambridge University Press:  23 October 2025

Li-Mei Dou ,
Hao Tang  and
Larry Wang
Li-Mei Dou
Affiliation:
Center for Combinatorics, LPMC, Nankai University, Tianjin, China
Hao Tang
Affiliation:
Center for Combinatorics, LPMC, Nankai University, Tianjin, China
Larry Wang*
Affiliation:
Center for Combinatorics, LPMC, Nankai University, Tianjin, China
*
Corresponding author: Larry Wang, email: wsw82@nankai.edu.cn
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Abstract

In this paper, we will investigate the following inequality

\begin{equation*}(a_{n+1}a_{n+2} - a_{n}a_{n+3})^2 -(a_{n+1}^2 -a_{n}a_{n+2})(a_{n+2}^2 - a_{n}a_{n+4}) \gt 0,\end{equation*}

which arises from the iterated Laguerre operator on functions. We will prove the sequence $\{a_n\}$ of a unified form given by Griffin, Ono, Rolen and Zagier asymptotically satisfies this inequality while the Maclaurin coefficients of the functions in Laguerre-Pólya class have not to possess this inequality. We also prove the companion version of this inequality. As a consequence, we show the Maclaurin coefficients of the Riemann Ξ-function asymptotically satisfy this property. Moreover, we make this approach effective and give the exact thresholds for the positivity of this inequalityfor the partition function, the overpartition function and the smallest part function.

Keywords

partition functionoverpartitionRiemann Ξ-functionTurán inequalityLaguerre inequality

MSC classification

Primary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Secondary: 11P82: Analytic theory of partitions 05A17: Partitions of integers

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.

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