论文标题
损坏的$ k $ -Diamond分区功能的不平等
Inequalities for the Broken $k$-Diamond Partition Function
论文作者
论文摘要
2007年，安德鲁斯（Andrews）和保勒（Paule）引入了损坏的$ k $ -diamond分区函数$δ_K（n）$，该功能已收到了有关算术属性的大量研究。在本文中，我们证明$ d^3 \logΔ_1（n-1）> 0 $ for $ n \ geq 5 $和$ d^3 \ logδ_2（n-1）> 0 $ for $ n \ geq 7 $，其中$ d $是差异操作员相对于$ n $。我们还猜想，对于任何$ k \ geq 1 $和$ r \ geq 1 $，都存在一个正整数$ n_k（r）$，因此对于$ n \ geq n_ {k} {k}（r）$，$（-1）这类似于分区函数对数有限差异的积极性，该差异已由Chen，Wang和XIE证明。此外，我们得到$ \ {δ_1（n）\} _ {n \ geq 0} $和$ \ {Δ_2（n）_ {n \ geq 0} $满足$ n \ geq 6 $的高级turán不平等。
In 2007, Andrews and Paule introduced the broken $k$-diamond partition function $Δ_k(n)$, which has received a lot of researches on the arithmetic propertises. In this paper, we prove that $D^3\log Δ_1(n-1)>0$ for $n\geq 5$ and $D^3 \log Δ_2(n-1)>0$ for $n\geq 7$, where $D$ is the difference operator with respect to $n$. We also conjecture that for any $k\geq 1$ and $r\geq 1$, there exists a positive integer $n_k(r)$ such that for $n\geq n_{k}(r)$, $(-1)^r D^r \log Δ_k(n)>0$. 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 $\{Δ_1(n)\}_{n\geq 0}$ and $\{Δ_2(n)\}_{n\geq 0}$ satisfy the higher order Turán inequalities for $n \geq 6$.
