Turán Inequalities for Infinite Product Generating Functions

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 $\text{func}pp(n)$ 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 $$0\le g_d\left(n\right)-n^d\le g_1\left(n\right){(n-1)}^{d-1}$$ 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 $\text{func}pp\left(n\right)=P_n^{{\sigma }_2}(1)$, where ${\sigma }_d(n):={\sum }_{\mathrm{\ell }\mid n}{\mathrm{\ell }}^d$ 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 $\text{func}id(n)=n$. Then $P_n^{\text{func}id}(x)=\frac{x}{n}{L}_{n-1}^{(1)}(-x)$, 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 ${\mathrm{\Delta }}_n^{g_d}(x)\ge 0$ for almost all $d$. Let $n\ge 6$ and $n\equiv ̸2(\mathrm{m}\mathrm{o}\mathrm{d}3)$. 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 $g_1=\text{func}id$ that ${\mathrm{\Delta }}_n^{\text{func}id}(x)\ge 0$ for all $x\ge 0$ and all $n$.