Hypergeometric function $_2F_1(1,1;c;z)$ for $z \leq -1$. The hypergeometric function $_2F_1(1,1;c;z)$, with $z < 0$ and $c \in (0,1)$. It can be written as a sum if $|z|  <1$:
$$
_2F_1(1,1;c;z) = \sum_{n=0}^{\infty} \frac{n!}{(c)_n} z^n,
$$
here $(c)_n$ is the Pochhammer Symbol.
This series does not converge for $|z| > 1$, what does this mean for $_2F_1(1,1;c;z)$? It does still exist as an integral but can it still be written as a (different) sum?
 A: There is a bunch of connection formulas relating hypergeometric functions at different arguments. A good source is $\S$ 15.10 (ii) Kummer’s 24 Solutions and Connection Formulas of Digital Library of Mathematical Functions.
In general, one can express
${}_2F_1(1,1;c;z)$ for nearly all $z \in \mathbb{C}$ using other hypergeometric
functions at argument where the defining series converges.
If I didn't make any mistake in using the formula there, following is what I found.
Please double check the formula yourself to ensure correctness!
When $\Re z < \frac12$, we have $\left|\frac{z}{z-1}\right| < 1$. We can use DLMF 15.10.11 to get
$${}_2F_1(1,1; c; z) = (1-z)^{-1}{}_2F_1\left(1,c-1;c;\frac{z}{z-1}\right)$$
When $\Re z > \frac12$ and $c \notin \mathbb{Z}$, we have $\left|\frac{z-1}{z}\right| < 1$. We can use DLMF 15.10.21 and the Euler's reflection formula $\Gamma(c)\Gamma(1-c) = \frac{\pi}{\sin(\pi c)}$ to get
$${}_2F_1(1,1; c; z) = \frac{c-1}{c-2} w_3(z)
+ \frac{\pi(1-c)}{\sin(\pi c)} w_4(z)
$$
where $w_3(z)$ is given by DLMF 15.10.13
$$w_3(z) = z^{-1} {}_2F_1\left(1,2-c;3-c; \frac{z-1}{z}\right)$$
and $w_4(z)$ is given by DLMF 15.10.14.
$$w_4(z) = z^{1-c}(1-z)^{c-2}{}_2F_1\left(0,c-1; c-1; \frac{z-1}{z}\right) = z^{1-c}(1-z)^{c-2}$$
