The Gaussian hypergeometric function is given by: $${}_2F_1[a,b,c,z]=\sum_{n=0}^{\infty}\dfrac{(a)_n(b)_n}{(c)_n n!}z^n,$$ where $(a)_n=a(a+1)\dots(a+n-1)$ for $n\in\mathbb{N}$ and $(a)_0=1$, is the Pochammer symbol.
If $|z|<1$, we easily obtain that ${}_2F_1[a,b,b,z]=(1-z)^{-a}$. If $|z|>1$, we can use the transformation: $$ {}_2F_1[a,b,c,z]=\dfrac{(-z)^{-a}\Gamma(c)\Gamma(a-c)}{\Gamma(b)\Gamma(b-a)} {}_2F_1[a,a-c+1,a-b+1,1/z]+\dfrac{(-z)^{-b}\Gamma(a-b)\Gamma(b)}{\Gamma(c-b)\Gamma(c)} {}_2F_1[a,b-c+1,b-a+1,1/z] $$ taking $c=b$ and keeping in mind that $\lim_{n\to0}\dfrac{1}{\Gamma(n)}=0.$ My question is, what to do in the case $|z|=1$? Or is there an analytic continuity argument to justify equality for any $z$?
Thanks a lot for the help.
The hypergeometric function $F (a, b; c; z)$ is defined by the Gauss series for $|z| < 1$ and by analytic continuation elsewhere (notably using Mellin-Barnes integrals; by using the hypergeometric transformation you mention for |z| > 1, you are actually performing a certain kind of analytic continuation; others exist).
In the special case $c=b$ you are mentioning, $(1-z)^{-a}$, when defined itself, is an analytic continuation of the Gauss hypergeometric function in the whole complex plane, except for $z\in [1,+\infty[$, which is a branch cut. You will have to choose a principal branch of the power function to work this out cleanly, unless $a\in\mathbb{Z}$, see NIST 4.2. So this simple continuation in particular works for $|z|>1, z\notin \mathbb{R}$ too without resorting to the hypergeometric transformation you mention in the post. Finally this transformation is only useful for real values of $z>1$.
Therefore there only remains to consider the special case $z=1$, for which two subcases arise.
Subcase 1
Either $a$ or $b$ is $0$ or a negative integer. Then the Gauss function reduces to a polynomial equal to the constant 1, if either $a=0$ or $b=0$, or of degree $|a|$ (resp. $|b|$), if $a$ (resp. $b$) is the only integer, or of degree $\min(|a|,|b|)$ if both are. In this case the value in 1 is trivially defined.
It so happens that this value has an elegant closed form. If $a=-n, n \in \mathbb{N}^*$, and $c$ is not $0$ or a negative integer, then $$F (−n, b; c; 1) = \frac{(c-b)_n}{(c)_n}$$(NIST 15.4.24), where $(\;)_n$ is the Pochhammer symbol, this formula being known as the Chu-Vandermonde identity. Note that when $b=c$,
$$F (−n, b; b; 1) = \frac{(0)_n}{(b)_n} = 0$$
a result that is not entirely trivial from the definition of the Gauss function as a power series.
Subcase 2
Neither $a$ nor $b$ is $0$ or a negative integer.
The Gauss hypergeometric function has a branch cut in the complex place in the interval $[1, +\infty[$. For $z=1$ its possibly defined value are well-known to be conditional on parameters $a,b, c$.
They are given by tabulated formulae in NIST and other such reference books:
$$\text{If }\; \Re(c − a − b) > 0, \text{then }\; F (a, b; c; 1) = \frac{Γ(c) Γ(c − a − b) }{Γ(c - a)Γ(c - b)}$$
(NIST 15.4.20) provided that $c-a$ and $c-b$ are not $0$ or negative integers (this formula is sometimes referred to as Gauss's hypergeometric theorem).
In particular when $c=b$ (or $c=a$), there is no defined value as $\Gamma(0)$ is not defined. You may however define it as a zero limit resulting from Gauss's theorem, provided that $\Re{a} < 0$ (otherwise there is a ratio of two infinities in Gauss's theorem). Mathematica and most software I know of follow this convention.
Otherwise values may only be defined as limits of equivalent functions when $z \to_{1-}$, check NIST 15.4.21-15.4.23.
Reference
National Institute of Standards and Technology (NIST) Handbook of Mathematical Functions, Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark, Cambridge University Press, 2010.