References for $\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$ and related integrals?

Question

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?

Answer 1

comment
$$ \binom{t}{1} = \frac{1}{\Gamma(t+1)\Gamma(2-t)} = \frac{\sin((t+1)\pi)}{t(t-1)\pi} $$ Using this, Maple does the indefinite integrals in terms of the functions ${\rm Si}$ and ${\rm Ci}$. Taking limits at $\pm \infty$ the first few results are $$ I\left( 1,1,1 \right) =2,\\ I\left( 1,1,2 \right) =2,\\ I\left( 1,1,3 \right) ={\frac{3}{2}}+6\,{\pi}^{-2},\\ I\left( 1,1,4 \right) ={\frac{4 }{3}}+10\,{\pi}^{-2},\\ I\left( 1,1,5 \right) ={\frac{115}{96}}+{\frac { 75}{8\,{\pi}^{2}}}+{\frac {105}{2\,{\pi}^{4}}},\\ I\left( 1,1,6 \right) ={\frac{11}{10}}+{\frac {21}{2\,{\pi}^{2}}}+{\frac {189}{2\,{\pi}^{4}} },\\ I\left( 1,1,7 \right) ={\frac{5887}{5760}}+{\frac {539}{48\,{\pi}^{ 2}}}+{\frac {735}{8\,{\pi}^{4}}}+{\frac {1155}{2\,{\pi}^{6}}}, \\I\left( 1,1,8 \right) ={\frac{302}{315}}+12\,{\pi}^{-2}+110\,{\pi}^{-4 }+{\frac {2145}{2\,{\pi}^{6}}},\\ I\left( 1,1,9 \right) ={\frac{259723}{ 286720}}+{\frac {13005}{1024\,{\pi}^{2}}}+{\frac {63855}{512\,{\pi}^{4 }}}+{\frac {135135}{128\,{\pi}^{6}}}+{\frac {225225}{32\,{\pi}^{8}}}, \\ I\left( 1,1,10 \right) ={\frac{15619}{18144}}+{\frac {1925}{144\,{\pi} ^{2}}}+{\frac {13585}{96\,{\pi}^{4}}}+{\frac {125125}{96\,{\pi}^{6}}}+ {\frac {425425}{32\,{\pi}^{8}}} $$