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