Asymptotics of a function

I have a function that looks like this:

$\displaystyle f=\sum_{p=0}^{n-1}{n-1 \choose p}(-1)^p \frac{(x-p)_n(y-p)_n}{(x+y-p)_n},$

where $\displaystyle (x)_n=x(x+1)\cdots (x+n-1).$

I want to let x and y go to infinity at the same rate. In order to do this, I let $\displaystyle z=x+y$ and $\displaystyle w=xy/z^2$. Then I let z go to infinity but keep w fixed.

I know, from a different calculation, that to leading order in z the function looks like

$\displaystyle f=n!zw\sum_{p=0}^{n-1}{n-1 \choose p}(-1)^p C_p w^p,$

where $\displaystyle C_p$ are the Catalan numbers.

However, I have not been able to prove this asymptotics directly from the definition of the function. Can anyone help me with this?? Thanks!