无穷乘积Generating Functions的Turán不等式
Turán Inequalities for Infinite Product Generating Functions
摘要：
在$1970$中，Nicolas证明了配分函数$p(n)$对于$ n > 25$是对数凹的。在\cite{HNT21}中，提出了一个关于$n >11$的平面配分函数$\func{pp}(n)$的对数对数正态性的精确猜想.最近，Ono、Pujahari和Rolen证明了这一点。在本文中，我们提供了一个总体情况。我们关联到双序列$\{g_d(n)\}_{d,n}$与$g_d(1)=1$和$$0 \leq g_{d}\left( n\right) - n^{d}\leq g_{1}\left( n\right) \left( n-1\right) ^{d-1}$$多项式$\{P_n^{g_d}(x)\}_{d,n}$由\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*}我们恢复$ p(n)= P_n^{σ_1}(1)$和$\func{pp}\left( n\right) = P_n^{σ_2}(1)$，其中$σ_d (n):= \sum_{\ell \mid n} \ell^d$和$f_d(n)= n^{d-1}$。让$n \geq 6$。则序列$\{P_n^{σ_d}(1)\}_d$对几乎所有$d$都是对数凹的当且仅当$n$可被$3$整除。让PDMATHBNG。然后是$P_n^{\func{id}}(x) = \frac{x}{n} L_{n-1}^{(1)}(-x)$，其中$L_{n}^{\left( α\right) }\left( x\right) $表示与$α$相关的拉盖尔多项式。本文研究了Turán不等式\begin{equation*} Δ_{n}^{g_d}(x) := \left( P_n^{g_d}(x) \right)^2 - P_{n-1}^{g_d}(x) \, P_{n+1}^{g_d}(x) \geq 0. \end{equation*}设$n \geq 6$和$0 \leq x < 2 - \frac{12}{n+4}$ .然后$n$可被$3$整除当且仅当$Δ_{n}^{g_d}(x) \geq 0$对于几乎所有的$d$。设$n \geq 6$和$n \not\equiv 2 \pmod{3}$。然后，$x$上的条件可以简化为$x \geq 0$。我们确定明确的界限。与Nicolas的结果类似，我们对$g_1= \func{id}$有$Δ_{n}^{\func{id}}(x) \geq 0$对所有$x \geq 0 $和所有$n$。
英文摘要：
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^{σ_1}(1)$ and $\func{pp}\left( n\right) = P_n^{σ_2}(1)$, where $σ_d (n):= \sum_{\ell \mid n} \ell^d$ and $f_d(n)= n^{d-1}$. Let $n \geq 6$. Then the sequence $\{P_n^{σ_d}(1)\}_d$ 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( α\right) }\left( x\right) $ denotes the $α$-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 $n \geq 6$ and $0 \leq x < 2 - \frac{12}{n+4}$. Then $n$ is divisible by $3$ if and only if $Δ_{n}^{g_d}(x) \geq 0$ 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 $Δ_{n}^{\func{id}}(x) \geq 0$ for all $x \geq 0 $ and all $n$.
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