Log-Concavity in Powers of Infinite Series Close to $(1-z{)}^{-1}$

Abstract

In this paper, we use the analytic method of Odlyzko and Richmond to study the log-concavity of power series. If $f(z)={\sum }_n{a}_n{z}^n$ is an infinite series with $a_n\ge 1$ 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 $\alpha <1$ such that $a_0+\cdots +a_n=C(n+1)-R_n$ where $0\le R_n\le O((n+1{)}^{\alpha })$, 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 $n$.

arXiv comment17 pages, no figures. Corrected many typos