TITLE: Gauss hypergeometric function at z=-1
QUESTION [5 upvotes]: is there anything like a special value case of the hypergeometric function if $z=-1$ such that one can evaluate $_2F_1(\alpha,\beta; \gamma; -1)$? I mean there is a nice representation for the case that we have $z=1$, but $z=-1$ seems to be a little bit problematic.

REPLY [3 votes]: The case $z=-1$ is not problematic, you can simply use e.g. the linear transformations
$$_2F_1\left( a,b;c;z \right)
= (1-z)^{-a} \,_2F_1\left(a,c-b;c;\frac{z}{z-1}\right)
= (1-z)^{-b} \,_2F_1\left(c-a,b;c;\frac{z}{z-1}\right)
$$
and get
$$_2F_1\left( a,b;c;-1 \right)
= 2^{-a} \, _2F_1\left(a,c-b;c;\frac{1}{2}\right)
= 2^{-b} \,_2F_1\left(c-a,b;c;\frac{1}{2}\right)
$$
And here a numerical example: $\quad _2F_1\left( 2,-5;1;-1 \right) = 112, \quad
_2F_1\left( 2,6;1;1/2 \right) = 448,$ as you can see they relatated by $\frac{1}{4} = 2^{-a}.$