Infinite sum over Gamma functions?

Question

I am having quite a bit of trouble understanding this sum. Can someone explain to me exactly how to this from 1 to 3,very easily way?

Question its from this webpage

Thanks.

Answer 1

$\color{red}{1}\mapsto \color{red}{2}$ is just a consequence of integration by parts. If $p(x)$ is a polynomial, a primitive of $p(x)\,e^{-x}$ is given by $\left[p(x)+p'(x)+p''(x)+\ldots\right]e^{-x}$. In our case we have $p(x)=\frac{x^{n+1}}{(n+1)!}$. $\color{red}{2}\mapsto \color{red}{3}$ comes from evaluating the given double series at $z=0$ and at $z=b$. In the first case (evaluation at $z=0$), just the contribute from $k=0$ might be different from zero.