Higher-dimensional box integrals. (English) Zbl 1251.33002

Borwein, Jonathan M.; Chan, O-Yeat; Crandall, R. E.

Exp. Math. 19, No. 4, 431-446 (2010).

Summary: Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of \(n\)-dimensional box integrals \(B_n(s) := \langle|\vec{r}|^s\rangle, \vec{r} \in [0, 1]^n\). In particular, we resolve an elusive integral called \(\mathcal{K}_5\) that previously acted as a “blockade” against closed-form evaluation in \(n = 5\) dimensions. In consequence, we now know that \(B_n\)(integer) can be given a closed form for \(n = 1, 2, 3, 4, 5\). We also find the general residue at the pole at \(s = - n\), this leading to new relations and definite integrals; for example, we are able to give the first nontrivial closed forms for six-dimensional box integrals and to show hyperclosure of \(B_6\)(even). The Clausen function and its generalizations play a central role in these higher-dimensional evaluations. Our results provide stringent test scenarios for symbolic-algebra simplification methods.

MSC:

33B30	Higher logarithm functions
11Y60	Evaluation of number-theoretic constants

Keywords:

box integral; trilogarithm; arctan integral

Software:

OEIS

References:

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