Tur\'an Inequalities for Infinite Product Generating Functions

TL;DR

This paper explores Turán inequalities for infinite product generating functions related to partition functions, establishing conditions under which these inequalities hold and connecting them to classical results on log-concavity.

Contribution

It introduces a general framework for Turán inequalities for a class of generating functions, extending classical results and providing explicit bounds and conditions.

Findings

Turán inequalities hold for almost all d when n is divisible by 3.

Explicit bounds for the variable x are provided for the inequalities to hold.

Connections are made between Turán inequalities and classical log-concavity results.

Abstract

In the $1970$s, Nicolas proved that the partition function $p(n)$ is log-concave for $ n > 25$. In \cite{HNT21}, a precise conjecture on the log-concavity for the plane partition function $\func{pp}(n)$ for $n >11$ was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences $\{g_d(n)\}_{d,n}$ with $g_d(1)=1$ and $$0 \leq g_{d}\left( n\right) - n^{d}\leq g_{1}\left( n\right) \left( n-1\right) ^{d-1}$$ polynomials $\{P_n^{g_d}(x)\}_{d,n}$ given by \begin{equation*} \sum_{n=0}^{\infty} P_n^{g_d}(x) \, q^n := \func{exp}\left( x \sum_{n=1}^{\infty} g_d(n) \frac{q^n}{n} \right) =\prod_{n=1}^{\infty} \left( 1 - q^n \right)^{-x f_d(n)}. \end{equation*} We recover $ p(n)= P_n^{\sigma_1}(1)$ and $\func{pp}\left( n\right) = P_n^{\sigma_2}(1)$, where $\sigma_d (n):= \sum_{\ell \mid n} \ell^d$ and $f_d(n)= n^{d-1}$. Let $n \geq 6$. Then the sequence $\{P_n^{\sigma_d}(1)\}_d$ is log-concave for almost all $d$ if and only if $n$ is divisible by $3$. Let $\func{id}(n)=n$. Then $P_n^{\func{id}}(x) = \frac{x}{n} L_{n-1}^{(1)}(-x)$, where $L_{n}^{\left( \alpha \right) }\left( x\right) $ denotes the $\alpha$-associated Laguerre polynomial. In this paper, we invest in Tur\'an inequalities \begin{equation*} \Delta_{n}^{g_d}(x) := \left( P_n^{g_d}(x) \right)^2 - P_{n-1}^{g_d}(x) \, P_{n+1}^{g_d}(x) \geq 0. \end{equation*} Let $n \geq 6$ and $0 \leq x < 2 - \frac{12}{n+4}$. Then $n$ is divisible by $3$ if and only if $\Delta_{n}^{g_d}(x) \geq 0$ for almost all $d$. Let $n \geq 6$ and $n \not\equiv 2 \pmod{3}$. Then the condition on $x$ can be reduced to $x \geq 0$. We determine explicit bounds. As an analogue to Nicolas' result, we have for $g_1= \func{id}$ that $\Delta_{n}^{\func{id}}(x) \geq 0$ for all $x \geq 0 $ and all $n$.