Asymptotics of Hypergeometric $_2F_1(a;b;c;z)$ for large $|z| \to \infty$?

Question

I found this list of asymptotics of the Gauss Hypergeometric function $_2F_1(a;b;c;z)$ here on Wolfram's site for large $|z| \to \infty$

In particular there is a general formula for $|z| \to \infty$ $$ _2F_1(a;b;c;z) \approx \frac{\Gamma(b-a)\Gamma(c)}{\Gamma(b)\Gamma(c-a)} (-z)^{-a} +\frac{\Gamma(a-b)\Gamma(c)}{\Gamma(a)\Gamma(c-b)} (-z)^{-b} $$ How is this derived? Also, is this always true (meaning, for all $a$, $b$, $c$)? There are no sources on the site I linked.

Is there also a way to determine the next-order terms?

Answer 1

It follows from the "reciprocation" formula $$ \eqalign{ & {}_2F_1 (a,b;c;z)\quad \left| \matrix{ \;a - b \notin Z \hfill \cr \;z \notin \left( {0,1} \right) \hfill \cr} \right.\quad = \cr & {{\Gamma (b - a)\Gamma (c)} \over {\Gamma (b)\Gamma (c - a)}}{1 \over {\left( { - z} \right)^{\,a} }} {}_2F_1 \left( {a,\,a - c + 1;\;a - b + 1;{1 \over z}} \right) + \cr & + {{\Gamma (a - b)\Gamma (c)} \over {\Gamma (a)\Gamma (c - b)}}{1 \over {\left( { - z} \right)^{\,b} }} {}_2F_1 \left( {b,\,b - c + 1;\;b - a + 1;{1 \over z}} \right) \cr} $$ (re., e.g., to this link )

That, in turn, is derived from the solutions of the hypergeometric differential equation.

Answer 2

Converting ${_2\hspace{-1px}F_1}$ to the Meijer G-function, we obtain $${_2\hspace{-1px}F_1}(a, b; c; z) = \frac {\Gamma(c)} {\Gamma(a) \Gamma(b)} G_{2, 2}^{1, 2} \left( -z \middle| {1 - a, 1 - b \atop 0, 1 - c} \right) = \\ \frac {\Gamma(c)} {2 \pi i \Gamma(a) \Gamma(b)} \int_{\mathcal L} \frac {\Gamma(y + a) \Gamma(y + b) \Gamma(-y)} {\Gamma(y + c)} (-z)^y dy.$$ A left loop is a valid contour for $|z| > 1$, and the sum of the residues at $y = -a - k$ and $y = -b - k$ gives a complete asymptotic expansion for large $|z|$. Excluding the logarithmic cases, $$\operatorname*{Res}_{y = -a - k} \frac {\Gamma(y + a) \Gamma(y + b) \Gamma(-y)} {\Gamma(y + c)} (-z)^y = \frac {(-1)^k \Gamma(b - a - k) \Gamma(a + k)} {\Gamma(c - a - k)} \frac {(-z)^{-a - k}} {k!}.$$