论文标题
Liu-Zhang不平等的多元化用于多阶层功能
Polynomization of the Liu-Zhang inequality for overpartition function
论文作者
论文摘要
令$ \ overline {p}（n）$表示分支机构功能。刘和张表明，通过使用Engle的分析结果，所有整数$ a，b> 1 $的$ \+叠加{我们在本文中提供了刘Zhang不平等的组合证明。 More precisely, motivated by the polynomials $P_{n}(x)$ , which generalize the $k$-colored partitions function $p_{-k}(n)$, we introduce the polynomials $\overline{P}_{n}(x)$, which take the number of $k$-colored overpartitions of $n$ as their special values. And by combining combinatorial and analytic approaches, we obtain that $\overline{P}_{a}(x) \overline{P}_{b}(x)>\overline{P}_{a+b}(x)$ for all positive integers $a,b$ and real numbers $x \ge 1$ , except for $（a，b，x）=（1,1,1），（2,1,1），（1,2,1）$。
Let $\overline{p}(n)$ denote the overpartition function. Liu and Zhang showed that $\overline{p}(a) \overline{p}(b)>\overline{p}(a+b)$ for all integers $a,b>1$ by using an analytic result of Engle. We offer in this paper a combinatorial proof to the Liu-Zhang inequaity. More precisely, motivated by the polynomials $P_{n}(x)$ , which generalize the $k$-colored partitions function $p_{-k}(n)$, we introduce the polynomials $\overline{P}_{n}(x)$, which take the number of $k$-colored overpartitions of $n$ as their special values. And by combining combinatorial and analytic approaches, we obtain that $\overline{P}_{a}(x) \overline{P}_{b}(x)>\overline{P}_{a+b}(x)$ for all positive integers $a,b$ and real numbers $x \ge 1$ , except for $(a,b,x)=(1,1,1),(2,1,1),(1,2,1)$.
