论文标题
无限产品生成功能的对数洞穴
Log-Concavity of Infinite Product Generating Functions
论文作者
论文摘要
在$ 1970 $ s中，尼古拉斯证明了生成函数定义的系数$ p_d（n）$。 q^n \ right）^{ - n^{d-1}}} \ end {equation*}是log-concave，对于$ d = 1 $。最近，Ono，Pujahari和Rolen将结果扩展到$ d = 2 $。请注意，$ p_1（n）= p（n）$是分区函数，$ p_2（n）= \ func {pp} \ left（n \ right）$是平面分区的数量。在本文中，我们以$ p_d（n）$为$ d $投资物业。令$ n \ geq 6 $。然后，$ p_d（n）$几乎是log-concave，$ n $可分配$ 3 $，否则几乎是严格的log-convex。
In the $1970$s Nicolas proved that the coefficients $p_d(n)$ defined by the generating function \begin{equation*} \sum_{n=0}^{\infty} p_d(n) \, q^n = \prod_{n=1}^{\infty} \left( 1- q^n\right)^{-n^{d-1}} \end{equation*} are log-concave for $d=1$. Recently, Ono, Pujahari, and Rolen have extended the result to $d=2$. Note that $p_1(n)=p(n)$ is the partition function and $p_2(n)=\func{pp}\left( n\right) $ is the number of plane partitions. In this paper, we invest in properties for $p_d(n)$ for general $d$. Let $n \geq 6$. Then $p_d(n)$ is almost log-concave for $n$ divisible by $3$ and almost strictly log-convex otherwise.
