Inequalities for the broken k-diamond partition functions

Abstract

In 2007, Andrews and Paule introduced the broken k-diamond partition function ${\Delta }_k\left(n\right)$. Many researches on the arithmetic properties for ${\Delta }_k\left(n\right)$ have been done. In this paper, we prove that $D^3\log {\Delta }_1(n-1)>0$ for $n\ge 5$ and $D^3\log {\Delta }_2(n-1)>0$ for $n\ge 7$, where D is the difference operator with respect to n. We also conjecture that for any $k\ge 1$ and $r\ge 1$, there exists a positive integer $n_k\left(r\right)$ such that for $n\ge n_k\left(r\right)$, ${(-1)}^r{D}^r\log {\Delta }_k\left(n\right)>0$. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been proved by Chen, Wang, and Xie. Furthermore, we obtain that both ${\left\{{\Delta }_1\right(n\left)\right\}}_{n\ge 0}$ and ${\left\{{\Delta }_2\right(n\left)\right\}}_{n\ge 0}$ satisfy the higher order Turán inequalities for $n\ge 6$.