Limit of hypergeometric function when first three parameters are all large I have encountered an interesting limit involving a particular parameterisation of the hypergeometric function.  The function of interest to me uses the parameters $1 \leqslant k \leqslant n \leqslant N$ and is given by:
$$H(N,n,k,z) \equiv \ _2F_1(n-N,k,n+1,z).$$
I am trying to find and prove the limit of this function as we take $N \rightarrow \infty$, $n \rightarrow \infty$ and $k \rightarrow \infty$ with the fixed limiting ratios $n/N \rightarrow \phi \in (0,1)$ and $k/n \rightarrow \lambda \in (0,1)$.  I would like to find a general form for the limiting function (which is a function of $z,\phi,\lambda$) if that is possible, but I am particularly interested in the limit of this function in a neighbourhood of $z=0$.  I think the limit reduces to an exponential function, but my reasoning is presently heuristic (see my answer below for my own heuristic reasoning).
My Question: What is the limiting function (which will be a function of $z,\phi,\lambda$) under the stipulated limit?  What is the best way to prove this limit formally?
 A: Here is what I have come up with heuristically, which is as far as  have got.  Taking the asymptotic equivalence $n \simeq \phi N$ and $k \simeq \lambda \phi N$ and taking $N$ to be "large" we get the asymptotic equivalence:
$$\begin{align}
H(N,n,k,z) 
&= \ _2F_1(n-N,k,n+1,z) \\[12pt]
&= \sum_{s=0}^\infty \frac{(n-N)_s (k)_s}{(n+1)_s} \frac{z^s}{s!} \\[6pt]
&\simeq \sum_{s=0}^\infty \frac{(-(1-\phi)N)_s (\lambda \phi N)_s}{(\phi N + 1)_s} \frac{z^s}{s!} \\[6pt]
&\simeq \sum_{s=0}^\infty \frac{(-(1-\phi)N)^s (\lambda \phi N)^s}{(\phi N + 1)^s} \frac{z^s}{s!} \\[6pt]
&\simeq \sum_{s=0}^\infty (-(1-\phi)N)^s \lambda^s \frac{z^s}{s!} \\[6pt]
&= \sum_{s=0}^\infty \frac{[-\lambda(1-\phi)Nz]^s}{s!} \\[12pt]
&= \exp(-\lambda(1-\phi)Nz), \\[6pt]
\end{align}$$
which would then be the limiting function.  Heuristically this seems okay to me, but the fourth step involves an implicit interchange of limits by bringing the limit inside the infinite sum.  Usually I would use Tannery's theorem for this, but I'm not clear on how to do this in the present context.  The heuristics look okay to me, but I'm not sure of the "order" of the approximation or how to prove the resulting limit formally.
