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arXiv · 2206.09512

Tur\'an inequalities for the broken $k$-diamond partition function

Janet J. W. Dong·Kathy Q. Ji·Dennis X. Q. Jia
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Abstract

We obtain an asymptotic formula for Andrews and Paule's broken $k$-diamond partition function $\Delta_k(n)$ where $k=1$ or $2$. Based on this asymptotic formula, we derive that $\Delta_k(n)$ satisfies the order $d$ Tur\'an inequalities for $d\geq 1$ and for sufficiently large $n$ when $k=1$ and $ 2$ by using a general result of Griffin, Ono, Rolen and Zagier. We also show that Andrews and Paule's broken $k$-diamond partition function $\Delta_k(n)$ is log-concave for $n\geq 1$ when $k=1$ and $2$. This leads to $\Delta_k(a)\Delta_k(b)\ge\Delta_k(a+b)$ for $a,b\ge 1$ when $k=1$ and $ 2$.

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Janet J. W. Dong, Kathy Q. Ji, Dennis X. Q. Jia. 2022-06-20. Tur\'an inequalities for the broken $k$-diamond partition function. https://arxiv.org/abs/2206.09512

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