Turán Inequalities for Infinite Product Generating Functions

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 $$0\le g_d\left(n\right)-n^d\le g_1\left(n\right){\left(n-1\right)}^{d-1}$$ 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 $\func{pp}\left( n\right) = P_n^{σ_2}(1)$, where ${\sigma }_d(n):={\sum }_{\ell \mid n}{\ell }^d$ and $f_d(n)=n^{d-1}$. Let $n\ge 6$. 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 Turán inequalities \begin{equation*} Δ_{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 $n\ge 6$ and $0\le x<2-\frac{12}{n+4}$. Then $n$ is divisible by $3$ if and only if ${\Delta }_n^{g_d}(x)\ge 0$ for almost all $d$. Let $n\ge 6$ and $n\not\equiv 2(\mathrm{mod}3)$. Then the condition on $x$ can be reduced to $x\ge 0$. We determine explicit bounds. As an analogue to Nicolas' result, we have for $g_1= \func{id}$ that $Δ_{n}^{\func{id}}(x) \geq 0$ for all $x\ge 0$ and all $n$.