论文标题
无限系列的log-concavity接近$（1-z）^{ - 1} $
Log-Concavity in Powers of Infinite Series Close to $(1-z)^{-1}$
论文作者
论文摘要
在本文中，我们使用Odlyzko和Richmond的分析方法来研究功率序列的对数。如果$ f（z）= \ sum_n a_nz^n $是一个无限系列，$ a_n \ geq 1 $和$ a_0 + \ cdots + a_n = o（n + 1）$（n + 1）$（n + 1）$，我们证明，超polynomemalisty上的长度很长的初始初始部分为$ f^k（z）$是log-concove。此外，如果存在常数，$ c> 1 $和$α<1 $，这样，$ a_0 + \ cdots + a_n = c（n + 1）-r_n $其中$ 0 \ leq r_n \ leq r_n \ leq o（（n + 1）^α）$，我们表明，这是一个$ f^k（z k（z）$ cog的初始初始段。这解决了登洪和作者提出的猜想，这意味着Heim和Neuhauser的另一种猜想是，Nekrasov-Okounkov多项式$ q_n（z）$是单峰，对于足够大的$ n $。
In this paper, we use the analytic method of Odlyzko and Richmond to study the log-concavity of power series. If $f(z) = \sum_n a_nz^n$ is an infinite series with $a_n \geq 1$ and $a_0 + \cdots + a_n = O(n + 1)$ for all $n$, we prove that a super-polynomially long initial segment of $f^k(z)$ is log-concave. Furthermore, if there exists constants $C > 1$ and $α< 1$ such that $a_0 + \cdots + a_n = C(n + 1) - R_n$ where $0 \leq R_n \leq O((n + 1)^α)$, we show that an exponentially long initial segment of $f^k(z)$ is log-concave. This resolves a conjecture proposed by Letong Hong and the author, which implies another conjecture of Heim and Neuhauser that the Nekrasov-Okounkov polynomials $Q_n(z)$ are unimodal for sufficiently large $n$.
