pr.probability - method of moments and Laplace transform from Shepp and Lloyd

Again from the Shepp and Lloyd paper "ordered cycle lengths in a random permutation", I found this puzzling equality. This one might require access to the paper itself since it's quite a mouthful:

In equation (15), they claimed it is straightforward that if there is an $F_r$ such that

$$int_0^1 exp(-y/xi) dF_r(xi) = int_y^{infty} frac{E(x)^{r-1}}{(r-1)!} frac{exp(-E(x) -x)}{x} dx $$

then $F_r$ will have moments $G_{r,m}$.

Here

$$G_{r,m} = int_0^{infty} frac{x^{m-1}}{m!} frac{E(x)^{r-1}}{(r-1)!} exp(-E(x)-x) dx$$

and

$$E(x) = int_x^{infty} frac{e^{-y}}{y} dy$$

which is related to the thread Reference request for a "well-known identity" in a paper of Shepp and Lloyd

It looks to me like some sort of Laplace transform, but I can't manage to get the algebra to work, because of the inverse exponent $y/xi$ with respect to $xi$.

I will be happy enough if one can tell me why we are looking at the transform $int_0^1 exp(-y/xi) dF_r(xi)$ instead of the usual moment generating function $int_0^1 exp(-y xi) dF_r(xi)$, or maybe it's a typo?