In an article they swapped the signs $\int$ and $\sum$ as follows
\begin{align*} S_{n}(\xi,x)&= (-1)^{n+1}\Big(\frac{d}{dx}\Big)^{n}\Big[\sum^{\infty}_{j=0}\int^{\infty}_{0}e^{-(j-n-\xi)t}L_{j}(x)dt\Big], \quad Re(\xi)<-n\\&=(-1)^{n+1}\Big(\frac{d}{dx}\Big)^{n} \int^{\infty}_{0}\Big[ \sum^{\infty}_{j=0}e^{-(j-n-\xi)t}L_{j}(x)dt\Big] \end{align*}
where $(L_{j})$ are the Laguerre polynomials of order zero.
See page 5 of this article for the full reference/context.
Who can tell me why?