Asymptotics of Hypergeometric $_2F_1(a;b;c;z)$ for large $|z| \to \infty$? 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?
 A: 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.
A: 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,
$$\operatorname*{Res}_{y = -a - k} \frac
 {\Gamma(y + a) \Gamma(y + b) \Gamma(-y)}
 {\Gamma(y + c)} (-z)^y =
\frac {(-1)^k \Gamma(b - a - k) \Gamma(a + k)} {\Gamma(c - a - k)}
 \frac {(-z)^{-a - k}} {k!}.$$
