Transforming a multiple intergral The following equality seems to hold:
\begin{equation}
\int_0^\infty\cdots\int_0^\infty F(\sum_{i=1}^kx_i)\text{ d}x_k\cdots\text{ d}x_1=\int_0^\infty \frac{y^{k-1}}{(k-1)!}F(y)\text{ d}y.
\end{equation}
Is there an easy way to see this?
 A: As Daniel Fischer suggested, let's do the change of variables:
$$
     x_k = y \cdot u_k
$$
where $0 < u_k<1$ such that $\sum_{k=1}^n u_k = 1$ and $y > 0$. Then Jacobian of this change of variables is 
$$
    J = y^{n-1}
$$
Hence
$$
    \int_0^\infty \cdots \int_0^\infty F\left(\sum_{k=1}^n x_k\right) \mathrm{d}x_1 \cdots \mathrm{d}x_n = \int_0^\infty y^{n-1} F\left(y \right) \mathrm{d}y \cdot \underbrace{\int_{u_k > 0, u_1+\cdots+u_{n-1} \leqslant 1} \mathrm{d}u_1 \cdots \mathrm{d}u_{n-1}}_{\Omega_{n-1}}
$$
In order to evaluate $\Omega_{n-1}$ it is best to proceed by recursion. By making another change of variables:
$$
   u_{n-1} = 1 - v_{n-1} \quad u_{k} = v_{n-1} v_{k} \quad \text{for } 1 \leqslant k < n-1
$$
with Jacobian $J = \left(v_{n-1}\right)^{n-2}$ we have
$$
   \Omega_{n-1} = \underbrace{\int_0^1 v_{n-1}^{n-2} \mathrm{d} v_{n-1}}_{\frac{1}{n-1}} \cdot \underbrace{\int_{v_k >0, v_1+\cdots+v_{n-2} < 1} \mathrm{d}v_1 \cdots \mathrm{d}v_{n-2}}_{\Omega_{n-2}}
$$
Hence 
$$
    \Omega_{n-1} = \frac{1}{n-1} \Omega_{n-2}
$$
which is easy to solve:
$$
   \Omega_{n-1} = \frac{1}{n-1}\cdot \frac{1}{n-2} \cdots \frac{1}{2} \Omega_1 = \frac{1}{(n-1)!}
$$
A: $\displaystyle{\large%
\int_{0}^{\infty}\cdots\int_{0}^{\infty}
{\rm F}\left(\sum_{i = 1}^{k}x_{i}\right){\rm d}x_{k}\ldots
{\rm d}x_{1}
=
\int_{0}^{\infty}{y^{k - 1} \over \left(k - 1\right)!}{\rm F}\left(y\right)\,
{\rm d}y
}$
$$------------------------------------$$
\begin{align}
&
\int_{0}^{\infty}\cdots\int_{0}^{\infty}
{\rm F}\,\left(\sum_{i = 1}^{k}x_{i}\right){\rm d}x_{k}\ldots
{\rm d}x_{1}
=
\int_{0}^{\infty}\cdots\int_{0}^{\infty}
\int_{\gamma - {\rm i}\infty}^{\gamma + {\rm i}\infty}
{{\rm d}s \over 2\pi{\rm i}}\,
\tilde{\rm F}\left(s\right)
\exp\left(s\sum_{i = 1}^{k}x_{i}\right)\,{\rm d}x_{k}\ldots{\rm d}x_{1}
\\[3mm]&=
\int_{\gamma - {\rm i}\infty}^{\gamma + {\rm i}\infty}
{{\rm d}s \over 2\pi{\rm i}}\,
\tilde{\rm F}\left(s\right)
\left[%
\int_{0}^{\infty}\exp\left(sx\right)\,{\rm d}x
\right]^{k}
=
\int_{\gamma - {\rm i}\infty}^{\gamma + {\rm i}\infty}
{{\rm d}s \over 2\pi{\rm i}}\,\tilde{\rm F}\left(s\right)
\,{\left(-1\right)^{k + 1} \over s^{k}}
\\[3mm]&=
\int_{\gamma - {\rm i}\infty}^{\gamma + {\rm i}\infty}
{{\rm d}s \over 2\pi{\rm i}}\,\int_{0}^{\infty}{\rm d}y\,{\rm e}^{-sy}\,
{\rm F}\left(y\right)\,{\left(-1\right)^{k + 1} \over s^{k}}
=
\int_{0}^{\infty}{\rm d}y\,{\rm F}\left(y\right)\left(-1\right)^{k + 1}
\int_{\gamma - {\rm i}\infty}^{\gamma + {\rm i}\infty}
{{\rm d}s \over 2\pi{\rm i}}\,
{{\rm e}^{-sy} \over s^{k}}
\\[3mm]&=
\int_{0}^{\infty}{\rm d}y\,{\rm F}\left(y\right)\left(-1\right)^{k + 1}
\int_{\gamma - {\rm i}\infty}^{\gamma + {\rm i}\infty}
{{\rm d}s \over 2\pi{\rm i}}\,
{1 \over s^{k}}
\sum_{n = 0}^{\infty}{\left(-y\right)^{n} \over n!}\,s^{n}
\\[3mm]&=
\int_{0}^{\infty}{\rm d}y\,{\rm F}\left(y\right)\sum_{n = 0}^{\infty}
\left(-1\right)^{k + 1}{\left(-y\right)^{n} \over n!}\quad
\overbrace{\int_{\gamma - {\rm i}\infty}^{\gamma + {\rm i}\infty}
{{\rm d}s \over 2\pi{\rm i}}\,{1 \over s^{k - n}}}^{{\LARGE\delta}_{k - n,1}}
=
\color{#ff0000}{\large%
\int_{0}^{\infty}{y^{k - 1} \over \left(k - 1\right)!}\,{\rm F}\left(y\right)}
\end{align}
