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关于某些广义二项和的非整性

On the nonintegrality of certain generalized binomial sums

Bernd C. Kellner

备注：
20 pages, 3 tables, 3 figures, final revised version
发表信息：
Integers 23 (2023), Article A45, 1-24
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AI总结：

本研究针对几类广义二项和，运用$p$进方法探讨其在整数参数下的非整性，推导分母性质、参数不等式及例外情形，推广了近期受关注的相关猜想的已有结论。

摘要：

我们研究了若干类广义二项和$\mathcal{S}_{(r,n)}(\ell)$，并运用$p$进方法，在多种情形下讨论了其在整数参数$n,r \geq 1$和$\ell \in \mathbb{Z}$下取值的非整性。特别地，我们证明了$\mathcal{S}_{(r,n)}(\ell)$的分母的若干性质。若将其视为多项式，序列$(\mathcal{S}_{(r,n)}(x))_{n \geq 0}$构成一个Appell序列。特殊情形$\mathcal{S}_{(r,n)}(2)$可约化为和式$\sum_{k=0}^{n} \binom{n}{k} \frac{r}{r+k}$，该和式的取值非整性猜想近期受到多位学者的关注，目前仅少数情形得到证明。这些广义结果除其他结论外，还蕴含当$|\ell| \geq 2$为偶数时，若$\binom{r+n}{r}$为偶数则有$\mathcal{S}_{(r,n)}(\ell) \notin \mathbb{Z}$，例如$r$和$n$均为奇数。尽管存在$\mathcal{S}_{(r,n)}(\ell) \in \mathbb{Z}$的例外情形，但对于任意固定的$|\ell| \geq 2$，满足$n,r \geq 1$的$\mathcal{S}_{(r,n)}(\ell)$的“几乎所有”取值都是非整数。随后，我们还推导了使得$\mathcal{S}_{(r,n)}(\ell) \notin \mathbb{Z}$成立的参数之间的显式不等式，尤其针对$r \geq n$和$n > r \geq \frac{1}{5} n$对应的$\ell$的某些小值证明了该结论。作为补充，我们最后讨论了$\mathcal{S}_{(r,n)}(\ell) \in \mathbb{Z}$的例外情形。

英文摘要：

We consider certain generalized binomial sums $\mathcal{S}_{(r,n)}(\ell)$ and discuss the nonintegrality of their values for integral parameters $n,r \geq 1$ and $\ell \in \mathbb{Z}$ in several cases using $p$-adic methods. In particular, we show some properties of the denominator of $\mathcal{S}_{(r,n)}(\ell)$. Viewed as polynomials, the sequence $(\mathcal{S}_{(r,n)}(x))_{n \geq 0}$ forms an Appell sequence. The special case $\mathcal{S}_{(r,n)}(2)$ reduces to the sum $\sum_{k=0}^{n} \binom{n}{k} \frac{r}{r+k}$, which has recently received some attention from several authors regarding the conjectured nonintegrality of its values. So far, only a few cases have been proved. The generalized results imply, among other things, for even $|\ell| \geq 2$ that $\mathcal{S}_{(r,n)}(\ell) \notin \mathbb{Z}$ when $\binom{r+n}{r}$ is even, e.g., $r$ and $n$ are odd. Although there exist exceptions where $\mathcal{S}_{(r,n)}(\ell) \in \mathbb{Z}$, ``almost all'' values of $\mathcal{S}_{(r,n)}(\ell)$ for $n,r \geq 1$ are nonintegral for any fixed $|\ell| \geq 2$. Subsequently, we also derive explicit inequalities between the parameters for which $\mathcal{S}_{(r,n)}(\ell) \notin \mathbb{Z}$. Especially, this is shown for certain small values of $\ell$ for $r \geq n$ and $n > r \geq \frac{1}{5} n$. As a supplement, we finally discuss exceptional cases where $\mathcal{S}_{(r,n)}(\ell) \in \mathbb{Z}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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