Asymptotic behavior of the hypergeometric function I'm trying to understand the asymptotic behavior of the hypergeometric function
$$
_2F_1(a, b; c; z)
$$
at fixed argument $0 < z < 1$ when some of the parameters $a$, $b$ and $c$ are large.
There are two specific cases I'm interested in, but I only know the answer in one of them. Let me start with the solved example to set the stage:
1) Consider
$$
_2F_1(a + n, b + n; c + 2n; z)
$$
in the limit $n \to \infty$. In this case one can use the saddle-point method in the integral representation of the hypergeometric function to show that
$$
_2F_1(a + n, b + n; c + 2n; z) \approx (1-z)^{(c-a-b-1/2)/2}
\left( 2 \frac{1 - \sqrt{1-z}}{z} \right)^{2n + c - 1}
$$
This is the kind of result I'm after: it shows that the hypergeometric function grows at most like $2^{2n}$ when $|z|<1$.
2) Now consider the other case of interest
$$
_2F_1(a + n, b - n; c; z)
$$
It is similar to the first case in the sense that the parameters are "balanced" in $n$. But now one of these parameters goes negative at large $n$, and I can observe numerically that the function oscillates in $n$. I assume that there should be an asymptotic expression for the amplitude of these oscillations, but I am not able to get one.
I cannot make progress neither with the saddle-point method nor with a variety of hypergeometric identities. Does anyone have a solution, or at least a suggestion on how to attack the problem?
 A: The idea is to take a contour $\gamma$ which starts at $0$, goes around $1$ counterclockwise, ends at $0$ and does not enclose $1/z$:
$$f(\zeta) = \frac
  {\Gamma(a - c + n + 1) \Gamma(c)}
  {2 \pi i \hspace {1px} \Gamma(a + n)}
 \zeta^{a - 1} (\zeta - 1)^{c - a - 1} (1 - z \zeta)^{-b}, \\
\phi(\zeta) = \ln \zeta - \ln(\zeta - 1) + \ln(1 - z \zeta), \\
{_2 F_1}(a + n, b - n; c; z) =
 \int_\gamma f(\zeta) e^{n \phi(\zeta)} d\zeta.$$
Then we can apply the steepest descent method. The stationary points of $\phi$ are
$$\zeta_{1, 2} = 1 \pm i \sqrt {\frac 1 z - 1},
\quad 0 < z < 1.$$
For $n \to \infty$, we obtain
$${_2 F_1}(a + n, b - n; c; z) =
f(\zeta_2) \sqrt {-\frac {2 \pi} {\phi''(\zeta_2) n}} \, e^{\phi(\zeta_2) n} -
 f(\zeta_1) \sqrt {-\frac {2 \pi} {\phi''(\zeta_1) n}} \, e^{\phi(\zeta_1) n} +
 R_n$$
(we take the principal value for the power functions; the plus sign before the square root corresponds to going through $\zeta_2$ in the direction from left to right).
The real parts of $\phi(\zeta_1)$ and $\phi(\zeta_2)$ are zero and $\Gamma(a - c + n + 1)/\Gamma(a + n)$ grows as $n^{1 - c}$, so $R_n = o(n^{1/2 - c})$.
