On the nonintegrality of certain generalized binomial sums

TL;DR

This paper investigates the nonintegrality of generalized binomial sums using $p$-adic methods, establishing conditions under which these sums are non-integers and analyzing their properties and exceptions.

Contribution

It provides new results on the nonintegrality of generalized binomial sums, including explicit inequalities and analysis of special cases, extending previous conjectures and partial proofs.

Findings

Most values of $ ext{S}_{(r,n)}( ext{ell})$ are nonintegral for fixed $| ext{ell}| extgreater 2$

Nonintegrality holds when $inom{r+n}{r}$ is even, e.g., for odd $r$ and $n$

Explicit parameter inequalities determine nonintegrality regions

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