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arXiv 2302.13327math.COmath.NT
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无穷乘积与无穷和Generating Functions的对数凹性

Log-Concavity of Infinite Product and Infinite Sum Generating Functions

Bernhard Heim, Markus Neuhauser

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摘要：

我们扩大的意见安德鲁斯的重要性，无限的总和和产品的组合。令$\{g_d(n)\}_{d\geq 0,n \geq 1}$为双序列$σ_d(n)= \sum_{\ell \mid n} \ell^d$或$ψ_d(n)= n^d$。我们关联双序列$\left\{ p^{g_{d} }\left( n\right) \right\}$和$\left\{ q^{g_{d} }\left( n\right) \right\} $，定义为\begin{eqnarray*} \sum_{n=0}^{\infty} p^{g_{d} }\left( n\right) \, t^{n} & := & \prod_{n=1}^{\infty} \left( 1 - t^{n} \right)^{-\frac{ \sum_{\ell \mid n} μ(\ell) \, g_d(n/\ell) }{n} }, \\ \sum_{n=0}^{\infty} q^{g_{d} }\left( n\right) \, t^{n} & := & \frac{1}{1 - \sum_{n=1}^{\infty} g_d(n) \, t^{n} }. \end{eqnarray*}的系数。这些系数与$n$的分区数$\mathrm{p}\left( n\right) = p^{σ_{1 }}\left ( n\right) $、平面分区数$pp\left( n\right) = p^{σ_{2 }}\left( n\right) $和Fibonacci数$F_{2n} = q^{ψ_{1 }}\left( n\right) $有关。让$n \geq 3$和让$n \equiv 0 \pmod{3}$。然后，系数是对数凹的$n$几乎所有的$d$在指数和几何的情况下。在这两种情况下，如果$n \equiv 2 \pmod{3}$，则几乎所有$d$的系数都不是对数凹的。让$n\equiv 1 \pmod{3}$。然后，对数凹属性几乎对所有$d$都翻转。

英文摘要：

We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let $\{g_d(n)\}_{d\geq 0,n \geq 1}$ be the double sequences $σ_d(n)= \sum_{\ell \mid n} \ell^d$ or $ψ_d(n)= n^d$. We associate double sequences $\left\{ p^{g_{d} }\left( n\right) \right\}$ and $\left\{ q^{g_{d} }\left( n\right) \right\} $, defined as the coefficients of \begin{eqnarray*} \sum_{n=0}^{\infty} p^{g_{d} }\left( n\right) \, t^{n} & := & \prod_{n=1}^{\infty} \left( 1 - t^{n} \right)^{-\frac{ \sum_{\ell \mid n} μ(\ell) \, g_d(n/\ell) }{n} }, \\ \sum_{n=0}^{\infty} q^{g_{d} }\left( n\right) \, t^{n} & := & \frac{1}{1 - \sum_{n=1}^{\infty} g_d(n) \, t^{n} }. \end{eqnarray*} These coefficients are related to the number of partitions $\mathrm{p}\left( n\right) = p^{σ_{1 }}\left ( n\right) $, plane partitions $pp\left( n\right) = p^{σ_{2 }}\left( n\right) $ of $n$, and Fibonacci numbers $F_{2n} = q^{ψ_{1 }}\left( n\right) $. Let $n \geq 3$ and let $n \equiv 0 \pmod{3}$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential and geometric cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n \equiv 2 \pmod{3}$. Let $n\equiv 1 \pmod{3}$. Then the log-concave property flips for almost all $d$.

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