Some Remarks on Sinc Integrals and Their Connection with Com- Binatorics, Geometry and Probability

where sinc(x) := (sinx)/x for x 6= 0 and sinc(0) := 1. They use a version of the Parseval/Plancherel formula to prove that, subject to certain conditions on the parameters aj , the sequence I1, I2, . . . is decreasing; they also give an explicit evaluation of In by expanding the product of sines into a sum of cosines and then integrating by parts. As the Borweins observed, the integral (1.1) can be interpreted geometrically as the volume of an n-dimensional polyhedron obtained by cutting an n-dimensional hypercube by two (n− 1)-dimensional hyperplanes, and it is interesting to note that their formula is essentially a signed sum over the vertices of the hypercube. It may therefore be of interest to give an independent evaluation of (1.1) using methods from combinatorial geometry. The derivation is provided in section 3. In [2], related integrals arise in a probabilistic setting—namely the problem of determining the probability density of a sum of independent random variables uniformly distributed in different ranges. We employ this interpretation here to give an alternative proof of an elegant combinatorial identity related to the evaluation of In using methods from probability theory. See Proposition 1 below.