I have a function that looks like this:

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

where (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 z=x+y and 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

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

where 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!