In user dxdydz's answer to the question "Unexpected appearances of $\pi^{2}/6$", the identity $$\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$$ is mentioned.

Here, we employ a generalization of the binomial coefficients to real arguments: $$\binom{x}{y} := \frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)} = \frac{1}{(x+1)B(y+1,x-y+1)} ,$$ where $B(\cdot,\cdot)$ is the Beta function.

I hadn't seen an integral quite like this one before. It turns out Ramanujan did work on it - as dxdydz states, it comes up in both Part 1 (p. 302 - 304) and Part 2 (p. 225-227) of his Notebooks.

In the answer to a version of this question I asked on MSE, user Marco Cantarini points out that: $$I(n,\alpha,l) := \int_{\mathbb{R}}\dbinom{n}{\alpha x}^{\ell}dx=\sum_{k\in\mathbb{Z}}\dbinom{n}{\alpha k}^{\ell},\,0<\alpha\leq2/\ell,\,\ell\in\mathbb{N}$$ and that a proof can be found here on MO.

I wonder, though, if there are any articles or books that delve into such integrals involving binomial coefficients more elaborately. In particular, I'm interested in descriptions detailing how to find a closed form for $I(1,1,l)$ with $l\in\mathbb{Z}_{\geq1}$. Do you know any references?