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

TL;DR

This paper derives asymptotic formulas for the broken k-diamond partition function and proves it satisfies Turán inequalities and log-concavity for large n when k=1 or 2, revealing new inequalities and properties.

Contribution

It provides the first asymptotic formulas and establishes Turán inequalities and log-concavity for the broken k-diamond partition function for k=1,2.

Findings

$oxed{ ext{Asymptotic formula for } riangle_k(n)}$

$oxed{ ext{Satisfaction of Turán inequalities for large }n$

$oxed{ ext{Log-concavity of } riangle_k(n)$ for }n ext{ ≥ 1}

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$.