论文标题
关于某些广义二项式总和的不整合性
On the nonintegrality of certain generalized binomial sums
论文作者
论文摘要
我们考虑某些广义的二项式总和$ \ MATHCAL {s} _ {（r，n）}（\ ell）$，并讨论其值的不整合性的积分参数$ n，r \ geq 1 $和$ \ ell \ in \ Mathbb {z} $的$ n，r \ geq 1 $和$ \ ell \ in \ mathbb {z} $。特别是，我们显示了$ \ Mathcal {s} _ {（r，n）}（\ ell）$的分母的一些属性。被视为多项式，序列$（\ MATHCAL {s} _ {（r，n）}（x））_ {n \ geq 0} $形成一个上方序列。特殊情况$ \ MATHCAL {s} _ {（r，n）}（2）$将$减少到总和$ \ sum_ {k = 0}^{n} \ binom {n} {n} {k} \ frac {r} {r} {r} {r+k} $，最近从几个授权者那里受到了一些授权的注意。到目前为止，仅证明了少数案件。广义的结果暗示，除其他外，甚至是$ | \ ell | \ geq 2 $ tht $ \ mathcal {s} _ {（r，n）}（\ ell）\ notin \ mathbb {z} $当$ \ binom {r+n} {r+n} {r} {r} $偶数，例如，$ r $和$ n $是奇数的。 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 $。随后，我们还会在$ \ mathcal {s} _ {（r，n）}（\ ell）\ notin \ notin \ mathbb {z} $的参数之间提出显着的不等式。特别是，对于$ r \ geq n $和$ n> r \ geq \ frac {1} {5} n $的某些小值显示了这一点。作为补充，我们最终讨论了$ \ Mathcal {s} _ {（r，n）}（\ ell）\ in \ Mathbb {z} $的特殊情况。
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 $\mathcal{S}_{(r,n)}(\ell) \in \mathbb{Z}$.
