Box integrals

摘要

By a "box integral" we mean here an expectation where r runs over the unit n-cube, with (q) over right arrow and s fixed, explicitly: integral(1)(0) ... integral(1)(0) ((r(1) - q(1))(2) + ... + (r(n) -q(n))(2))(s/2) dr(1) ... 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. (c) 2006 Elsevier B.V. All rights reserved.