Lawrence Berkeley National Laboratory

By a "box integral" we mean here an expectation $\langle |\vec r - \vec q|^s \rangle$ where $\vec r$runs over the unit $n$-cube, with $\vec q$ and $s$ fixed, explicitly:\begin eqnarray* &&\int_01 \cdots \int_01 \left((r_1 - q_1)2 + \dots +(r_n-q_n)2\right)^ s/2 \ dr_1 \cdots dr_n.\end eqnarray* 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.