A Darling-Siegert formula relating some Bessel integrals and random walks (Q2373653)

The authors discuss in detail the formula \[ \sum_{k=0}^{r}\Big({2k+m \atop k+m}\Big)\,\Big({2r-2k+m \atop r-k+m}\Big)\,\frac{1}{k+m+1}=\frac{1}{m+1}\,\Big({2r+2m+1 \atop r+2m+1}\Big), \eqno (1) \] for \(m=0,1,\ldots\) which appears in the study of random flights in the space \(\mathbb{R}^4\). They give a probabilistic interpretation of this formula and provide a combinatorial proof of it. Several other results concerning sums as in the left hand side of (1) are given.