I have been trying to determine the series expansion of the beta function, but so far I haven't been successful. The two results I wish to obtain are the following:
$$ B(x,y) = \sum_{n=0}^{\infty} \frac{\binom{n-y}{n} }{x+n} $$
and \begin{equation} B(x,y) = \sum_{n=0}^{\infty} \frac{1}{n! (n+x)} \frac{\Gamma(n-y+1)}{\Gamma(1- y)} \end{equation}
For the second result, it follows directly from (Eq. 16.79 of Basic Concepts of String Theory)
$$B(x,y) = \frac{1}{\Gamma(1-y)} \int_{0}^{\infty} ds \int_0^1 dz \; s^{-y} e^{-(1-z)s} z^{x-1} $$
However, I don't understand what were the steps to obtain this equality.
Therefore, any hints on how to proceed in obtaining the first result and this last equation would be highly appreciated.