On the nonintegrality of certain generalized binomial sums

Abstract

We consider certain generalized binomial sums ${\mathcal{S}}_{(r,n)}(\ell )$ and discuss the nonintegrality of their values for integral parameters $n,r\ge 1$ 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 $({\mathcal{S}}_{(r,n)}(x){)}_{n\ge 0}$ forms an Appell sequence. The special case ${\mathcal{S}}_{(r,n)}(2)$ reduces to the sum ${\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\frac{r}{r+k}$, 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 $|\ell |\ge 2$ that ${\mathcal{S}}_{(r,n)}(\ell )\notin \mathbb{Z}$ when $\left(\genfrac{}{}{0ex}{}{r+n}{r}\right)$ 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 $n,r\ge 1$ are nonintegral for any fixed $|\ell |\ge 2$. 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 $r\ge n$ and $n>r\ge \frac{1}{5}n$. As a supplement, we finally discuss exceptional cases where ${\mathcal{S}}_{(r,n)}(\ell )\in \mathbb{Z}$.

20 pages, 3 tables, 3 figures, final revised version