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

TL;DR

This paper proves that powers of certain infinite series close to (1-z)^{-1} exhibit long initial segments of log-concavity, confirming conjectures related to polynomial unimodality.

Contribution

It establishes conditions under which powers of specific infinite series have long segments of log-concavity, resolving related conjectures.

Findings

Super-polynomially long initial segments are log-concave.

Exponentially long initial segments are log-concave under specific conditions.

Confirms conjecture that Nekrasov-Okounkov polynomials are unimodal for large n.

Abstract

In this paper, we use the analytic method of Odlyzko and Richmond to study the log-concavity of power series. If $f(z) = \sum_n a_nz^n$ 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 $\alpha < 1$ such that $a_0 + \cdots + a_n = C(n + 1) - R_n$ where $0 \leq R_n \leq O((n + 1)^{\alpha})$, 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 $n$.