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 $\pi^{2}/6$", he or she mentions the identity $$\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}.$$ 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.

I wonder, though, if there are any articles or books that delve into such integrals involving binomial coefficients more elaborately. Do you know any references?

Answer 1

In general we have that $$\int_{\mathbb{R}}\dbinom{n}{\alpha x}^{\ell}dx=\sum_{k\in\mathbb{Z}}\dbinom{n}{\alpha k}^{\ell},\,0<\alpha\leq2/\ell,\,\ell\in\mathbb{N}$$ and you can find a proof of this formula here. It is a special case of the following result of Robert Baillie, David Borwein and Jonathan M. Borwein:

Theorem. Assume that $G$ is of bounded variation on $[-\delta,\delta]$, vanishes outside $\left(-\alpha,\alpha\right)$ and it is Lebesgue integrable over $\left(-\alpha,\alpha\right)$ with $0<\alpha<2\pi$ and define its Fourier transform as $$g\left(x\right):=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}G\left(u\right)e^{-iux}du.$$ Then $$\lim_{r\rightarrow+\infty}\sum_{k=-r}^{r}g\left(n\right)=\lim_{T\rightarrow+\infty}\int_{-T}^{T}g\left(x\right)dx=\sqrt{\frac{\pi}{2}}\left(G\left(0^{+}\right)+G\left(0^{-}\right)\right)$$

For a reference see here, Proposition $2$. Your particular integral is not covered by the previous theorem but maybe it is possible to adapt the ideas in the article for the sinc function case.