I have a function that looks like this:

where

I want to let x and y go to infinity at the same rate. In order to do this, I let and . 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

where 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!