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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 deftly 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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  • $\begingroup$ 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. $\endgroup$
    – Gjergji Zaimi
    Jul 28, 2011 at 10:01
  • $\begingroup$ 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? $\endgroup$
    – Robby McKilliam
    Jul 28, 2011 at 11:36

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