# More multinomial type integrals over the hypercube

The question is related to my previous question about integrating the multinomial over the hypercube and the motivation for this question is the same, but the integral is a bit different. Here it is,

$$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i^2\right)^k \left( \sum_{i=1}^{n}x_i\right)^m dx_1d x_2\dots dx_n.$$

Richard Stanley defltly solved the last integral, but this one does not appear to accept the same approach (the $x_i^2$ gets in the way).

I think I might be able to solve it using the partition function, but the solution is going to be terribly ugly. I am hoping that someone knows a tidy solution, or knows good references for solving integrals of this type.

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Of course by the same argument you only need to compute the corresponding integrals for $n=1$, and then raise the bivariate exponential generating function to the $n$th power. The integral of $\exp(tx^2+sx)$ is not very nice though, but at least it gives you an efficient way to compute the answer. – Gjergji Zaimi Jul 28 '11 at 10:01
As I said the $x_i^2$ gets in the way. If is was just two multinomials, one to power $k$, the other to power $m$, there would be no problem, you would get $\exp(tx + sx)$ and everything would work out nicely as before. Perhaps I have missed something though. How do you intend to use the integral of $\exp(tx^2 + sx)$ (which has no closed form solution as far as I am aware) taken to the power of $n$ to efficiently compute the answer? – Robby McKilliam Jul 28 '11 at 11:36