Asymptotic behavior of the hypergeometric function

Question

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?

Answer 1

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 $R_n = o(n^{1/2 - c})$.