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arXiv 1407.0177math.NTmath.CO
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分拆函数的对数的Finite Differences

Finite Differences of the Logarithm of the Partition Function

William Y. C. Chen, Larry X. W. Wang, Gary Y. B. Xie

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28 pages
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摘要：

令$p(n)$表示分拆函数。DeSalvo和Pak证明了$\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)> \frac{p(n)}{p(n+1)}$对于$n\geq 2$，正如Chen所证明的。此外，他们证明了一个更尖锐的不等式$\frac{p(n-1)}{p(n)}\left( 1+\fracπ{\sqrt{24}n^{3/2}}\right) > \frac{p(n)}{p(n+1)}$对$n\geq 45$成立。本文通过给出$-Δ^{2} \log p(n-1)$的一个上界证明了Desalvo和Pak的猜想，其中$Δ$是关于$n$的差分算子.我们还表明，对于给定的$r\geq 1$和足够大的$n$，$(-1)^{r-1}Δ^{r} \log p(n)>0$。这类似于分拆函数的有限差分的正性。Good提出并由Gupta证明，对于给定的$r\geq 1$，对于足够大的$n$，$Δ^{r} p(n)>0$。

英文摘要：

Let $p(n)$ denote the partition function. DeSalvo and Pak proved that $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)> \frac{p(n)}{p(n+1)}$ for $n\geq 2$, as conjectured by Chen. Moreover, they conjectured that a sharper inequality $\frac{p(n-1)}{p(n)}\left( 1+\fracπ{\sqrt{24}n^{3/2}}\right) > \frac{p(n)}{p(n+1)}$ holds for $n\geq 45$. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for $-Δ^{2} \log p(n-1)$, where $Δ$ is the difference operator with respect to $n$. We also show that for given $r\geq 1$ and sufficiently large $n$, $(-1)^{r-1}Δ^{r} \log p(n)>0$. This is analogous to the positivity of finite differences of the partition function. It was conjectured by Good and proved by Gupta that for given $r\geq 1$, $Δ^{r} p(n)>0$ for sufficiently large $n$.

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