# Advances in the theory of box integrals

Bailey, David H. and Borwein, Jonathan M. and Crandall, Richard E. (2010) Advances in the theory of box integrals. Mathematics of Computation, 79 . pp. 1839-1866.

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Box integrals, being expectations such as $\langle |\vec r|^s \rangle$ or $\langle |\vec r - \vec q|^s \rangle$ with $\vec r, \vec q$ chosen randomly over the unit $n$-cube, have over the years been given closed forms for certain $n, s$. By employing experimental mathematics together with a new analytic strategy, we have been able to prove that for $n = 1,2,3$ dimensions, and any integer $s$, the box integrals are hypergeometrically closed," in a sense we clarify herein. We are also able to supply a compendium of new, exact box-integral evaluations that arise naturally from the theory.