Question

I have come across an integral of the form $$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i\right)^mdx_1d x_2\dots dx_n.$$ I have a solution that makes use of the partition function, but I feel there should be a much nicer solution and I'm sure this has been looked at before. Does anybody know a reference?

Motivation: This integral has appeared whilst trying to compute moments of the Voronoi cell of the lattice $A_n$ (see page 462 of Sphere Packings, Lattices and Groups)

Answer 1

We want the coefficient of $t^m/m!$ in $$ \int_b^a\cdots \int_b^a e^{(x_1+\cdots+x_n)t}dx_1\dots dx_n = \left(\frac{e^{at}-e^{bt}}{t}\right)^n. $$ Expanding by the binomial theorem and then taking the coefficient of $t^m/m!$ from each term will give a formula.