Advances in the theory of box integrals

Box integrals—expectations 〈|~r|〉 or 〈|~r − ~q|〉 over the unit n-cube—have over three decades been occasionally given closed forms for isolated n, s. By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of n = 1, 2, 3, 4 dimensions the box integrals are for any integer s hypergeometrically closed (“hyperclosed”) in an explicit sense we clarify herein. For n = 5 dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call K5— although we do prove that all but a finite set of (n = 5) cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory. Correction (17 Mar 2020): This update corrects equations (10), (14) and (15), to show summations starting with j = 0. ∗Lawrence Berkeley National Laboratory, Berkeley, CA 94720, dhbailey@lbl.gov. Supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC0205CH11231. †School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia jonathan.borwein@newcastle.edu.au and Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2W5, Canada, jborwein@cs.dal.ca. Supported in part by ARC, NSERC and the Canada Research Chair Programme. ‡Center for Advanced Computation, Reed College, Portland OR, crandall@reed.edu.