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+⋯+(rn-qn)2)s/2dr1…drn.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=1s=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.