Advances in the theory of box integrals

Authors:
D. H. Bailey, J. M. Borwein and R. E. Crandall

Journal:
Math. Comp. **79** (2010), 1839-1866

MSC (2010):
Primary 11Y60; Secondary 28-04

DOI:
https://doi.org/10.1090/S0025-5718-10-02338-0

Published electronically:
February 9, 2010

MathSciNet review:
2630017

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Abstract | References | Similar Articles | Additional Information

Abstract: Box integrals--expectations or over the unit -cube--have over three decades been occasionally given closed forms for isolated . By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of dimensions the box integrals are for any integer hypergeometrically closed (``hyperclosed'') in an explicit sense we clarify herein. For dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call ; although we do prove that all but a finite set of () cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory.

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Additional Information

**D. H. Bailey**

Affiliation:
Lawrence Berkeley National Laboratory, Berkeley, California 94720

Email:
dhbailey@lbl.gov

**J. M. Borwein**

Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia and Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2W5, Canada

Email:
jonathan.borwein@newcastle.edu.au, jborwein@cs.dal.ca

**R. E. Crandall**

Affiliation:
Center for Advanced Computation, Reed College, Portland, Oregon

Email:
crandall@reed.edu

DOI:
https://doi.org/10.1090/S0025-5718-10-02338-0

Received by editor(s):
March 3, 2009

Received by editor(s) in revised form:
August 13, 2009

Published electronically:
February 9, 2010

Additional Notes:
The first author was 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-AC02-05CH11231.

The second author was supported in part by ARC, NSERC and the Canada Research Chair Programme.

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.