I am currently implementing the 2F1 Gaussian hypergeometric function numerically, and need to know its continuation for $ |z| > 1 $.
I have researched this and found this nice formula in the wolfram alpha documentation for $ |z| > 1 $ and $ a-b \notin \Bbb Z$ :
$_2F_1(a,b,c,z) = \frac {\Gamma(b-a) \Gamma(c) (-z)^{-a}}{\Gamma(b) \Gamma(c-a)}{\sum_{k=0}^{∞} \frac{(a)_k(a-c+1)_k z^{-k}}{k! (a-b+1)_k} }+\frac {\Gamma(a-b) \Gamma(c)(-z)^{-b}}{\Gamma(a) \Gamma(c-b)}{\sum_{k=0}^{∞} \frac{(b)_k(b-c+1)_k z^{-k}}{k! (-a+b+1)_k} }$
The problem I have with this formula is the $ a-b \notin \Bbb Z$ condition, because in the case I am interested in a-b is a natural number. One could now say that for $ a-b \in \Bbb Z$ there just is no formula outside the unit circle. However, Mathematica or Wolfram Alpha does actually return real values for this case, for example
$_2F_1(1,1,0.5,-1.2) = 0.136$
$_2F_1(1,2,0.5,-5.2) = -0.0974$
So my question is how does Mathematica/Wolfram Alpha compute these values for $ a-b \in \Bbb Z$? Is there an expression I am missing here or some documentation that shows me what formulae Mathematica uses?
If $z<0$ we can use transformation formulae to write $$ {_2F}_1\left({a,b\atop c};z\right)=(1-z)^{-b}{_2F}_1\left({c-a,b\atop c};\frac{z}{z-1}\right). $$ Notice that $0<z/(z-1)<1$ so we may us the series representation to compute the hypergeometric function on the right hand side.
If you need a continuation for other values of $z$ outside the unit disk when $b-a=0,1,2,\dots$ see DLMF 15.8.8.