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: ∫ 0 1 ⋯ ∫ 0 1 ( ( r 1 - q 1 ) 2 + ⋯ + ( r n - q n ) 2 ) s / 2 d r 1 … d r 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.