Advances in the theory of box integrals

Bailey, D.; Borwein, J.; Crandall, R.

doi:10.1090/S0025-5718-10-02338-0

Advances in the theory of box integrals
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by D. H. Bailey, J. M. Borwein and R. E. Crandall PDF

Math. Comp. 79 (2010), 1839-1866 Request permission

Abstract:

Box integrals—expectations $\langle |\vec r|^s \rangle$ or $\langle |\vec r - \vec q|^s \rangle$ 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 ${\mathcal K}_5$; 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.

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