论文标题
无限产品生成功能的Turán不平等现象
Turán Inequalities for Infinite Product Generating Functions
论文作者
论文摘要
在$ 1970 $ s中，尼古拉斯证明了分区功能$ p（n）$是$ n> 25美元的log-concave。在\ cite {hnt21}中，陈述了$ n> 11 $的平面分区函数$ \ func {pp}（n）$的log-concavity的精确猜想。最近，这是由Ono，Pujahari和Rolen证明的。在本文中，我们提供了一张一般图片。我们将$ \ {g_d（n）\} _ {d，n} $与$ g_d（1）= 1 $ and $$ 0 \ leq g_ {d} \ left（n \ right） - n ^{d} \ leq g_ g_ {1} {1} \ lest（n \左） polyenomials $ \ {p_n^{g_d}（x）\} _ {d，n} $由\ begin \ begin {equation*} \ sum_ {n = 0}^{\ infty} {\ infty} p_n^p_n^{g_d} {g_d} {g_d}（x） \ sum_ {n = 1}^{\ infty} g_d（n）\ frac {q^n} {n} {n} \ right）= \ prod_ {n = 1}^{\ infty} \ left（1- q^n \ right（1- q^n \ right）^{ - x f_d（n）}。 \ end {equation*}我们恢复$ p（n）= p_n^{σ_1}（1）$和$ \ func {pp} \ left（n \ oright）= p_n^{σ_2}（1）$ n^{d-1} $。令$ n \ geq 6 $。那么序列$ \ {p_n^{σ_d}（1）\} _ d $是log-concave，几乎是所有$ d $的log-concave，并且仅当$ n $可将$ n $除以$ 3 $。令$ \ func {id}（n）= n $。 Then $P_n^{\func{id}}(x) = \frac{x}{n} L_{n-1}^{(1)}(-x)$, where $L_{n}^{\left( α\right) }\left( x\right) $ denotes the $α$-associated Laguerre polynomial.在本文中，我们投资于Turán不平等\ begin {equation*}δ_{n}^{g_d}（x）（x）：= \ left（p_n^{g_d}（g_d}（x）\ right）^2 -p_ p_ p_ {n -1}} \ end {equation*}令$ n \ geq 6 $和$ 0 \ leq x <2- \ frac {12} {n+4} $。然后，$ n $在几乎所有$ d $的$δ__{n}^{g_d}（x）（x）\ geq 0 $的情况下，只有$ 3 $可将$ 3 $排除。令$ n \ geq 6 $和$ n \ not \ equiv 2 \ pmod {3} $。然后，$ x $的条件可以将其简化为$ x \ geq 0 $。我们确定明确的界限。作为对尼古拉斯结果的类比，我们有$ g_1 = \ func {id} $ that $δ_{n}^{\ func {id}}}}（x）\ geq 0 $ for ALL $ x \ geq 0 $和所有$ n $。
In the $1970$s, Nicolas proved that the partition function $p(n)$ is log-concave for $ n > 25$. In \cite{HNT21}, a precise conjecture on the log-concavity for the plane partition function $\func{pp}(n)$ for $n >11$ was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences $\{g_d(n)\}_{d,n}$ with $g_d(1)=1$ and $$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 $n$.
