Log-Concavity of Infinite Product Generating Functions

TL;DR

This paper investigates the log-concavity properties of coefficients from a family of infinite product generating functions, extending known results for specific cases to a general parameter d, and analyzing their behavior for large n.

Contribution

It generalizes the log-concavity results of the coefficients p_d(n) for all integer d, providing new insights into their asymptotic properties.

Findings

p_d(n) is almost log-concave for n divisible by 3

p_d(n) is almost strictly log-convex for n not divisible by 3

Results extend previous work from d=1,2 to general d

Abstract

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.