Advances in the theory of box integrals

Abstract

Box integrals-expectations | r| s or | r − q| s over the unit n-cube (or nbox)-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 n ≤ 4 dimensions the box integrals are for any integer s hypergeometrically closed in a sense we clarify herein. For n = 5 dimensions, we show that a single unresolved integral we call K 5 stands in the way of such hyperclosure proofs. We supply a compendium of exemplary closed forms that naturally arise algorithmically from this theory.

Exploratory Experimentation: Digitally-Assisted Discovery and Proof

“…This is equally true of extreme-precision calculation -a prerequisite for much of my own work [3,7] and others [9] -or in combinatorics.…”

“…Doubtless the greatest dissolvent in contemporary thought of old questions, the greatest precipitant of new methods, new intentions, new problems, is the one effected by the scientific revolution that found its climax in the "Origin of Species". "-John Dewey (1859-1952) 7 Thomas Kuhn has noted that a true paradigm shift -as opposed to the cliché -is "a conversion experience." 8 You (and enough others) either have one or you don't.…”

“…Moreover, the process of specifying my questions enough to program with them invariably enhances my understanding and sometimes renders the actual computer nearly superfluous. For example, in a recent study of expectation or "box integrals" [7] we were able to evaluate a quantity which had defeated us for years, namely…”