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 $n,r \geq 1$ 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 $\mathcal{S}_{(r,n)}(\ell) \in \mathbb{Z}$.