I am trying to compute a limit of a sum, and Mathematica can only write the sum as a Hypergeometric function, but it cannot compute the limit. So I have:
$$\lim_{N \rightarrow \infty} \frac{1}{ {2N \choose N }} \sum_{k=0}^N {N \choose k}^2 \, \left[ \frac{(1+z/N)(1+w/N)}{(1+u/N)(1+v/N)} \right]^k = \\ = \lim_{N \rightarrow \infty} \frac{1}{ {2N \choose N }} \; \ _2F_1 \left(-N, -N, 1, \frac{(1+z/N)(1+w/N)}{(1+u/N) (1+v/N)} \right)$$
where $z$, $w$, $u$ and $v$ are complex numbers. Any ideas on how to take the limit, using some trick on the summation or some properties of the Hypergeometric function?