Box integrals

By a “box integral” we mean here an expectation <|→/r − →/q|s> where →/r runs over the unit n-cube, with →/q and s fixed, explicitly: ∫₀¹...∫₀¹((r₁-q₁)²+...+(r_n-q_n)²)^s/2dr₁...dr_n. The study of box integrals leads one naturally into several disparate fields of analysis. While previous studies have focused upon symbolic evaluation and asymptotic analysis of special cases (notably s = 1), we work herein more generally—in interdisciplinary fashion—developing results such as: (1) analytic continuation (in complex s), (2) relevant combinatorial identities, (3) rapidly converging series, (4) statistical inferences, (5) connections to mathematical physics, and (6) extreme-precision quadrature techniques appropriate for these integrals. These intuitions and results open up avenues of experimental mathematics, with a view to new conjectures and theorems on integrals of this type.