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arXiv · 2312.10909

Laguerre inequalities and determinantal inequalities for the finite difference of the partition functions

Eve Y. Y. Yang
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Abstract

The paper aims to establish the Tur\'an inequalities, the Laguerre inequalities (order $2$), and the determinantal inequalities (order $3$) for $\Delta p(n)$ and $\Delta \bar{p}(n)$, where $\Delta f(n)$ is the first-order forward difference of a sequence $f(n)$. The functions $p(n)$ and $\bar{p}(n)$ denote the partition function and overpartition function, respectively. Conjectures for thresholds of Laguerre inequalities (order $m$) and positivity of $m$-order determinants are proposed, extending to $\Delta^k p(n)$ and $\Delta^k \bar{p}(n)$, with $1 \leq m \leq 11$ and $1 \leq k \leq 5$.

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Eve Y. Y. Yang. 2023-12-18. Laguerre inequalities and determinantal inequalities for the finite difference of the partition functions. https://arxiv.org/abs/2312.10909

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