I am looking for an expression of the asymptotic behaviour for $|z|\rightarrow\infty$ of the Hypergeometric function ${}_2F_1(a,b,c,z)$ when the condition $a-b \not\in {\mathbb Z}$ is not satisfied, in particular when $a=b$. Previous questions and answer, like here using the reciprocation formula, have not considered this case.
If I use the Euler Hypergeometric transformation, I could related the limit $|z|\rightarrow\infty$ to the limit $z\rightarrow 1$ as $${}_2F_1(a,b,c;z) = (1-z)^a {}_2F_1(a, c-b, c; z / (z-1))$$
Then for $|z|\rightarrow\infty$ the right-hand side gives $$\sim (1-z)^a {}_2F_1(a, c-b, c; 1) = (1-z)^a \frac{\Gamma(c)\Gamma(b-a)}{\Gamma(c-a)\Gamma(b)}$$
This expression seems correct, as long as the argument of the $\Gamma-$function does not hit a pole. But what will happen if $a=b$? What is then the asymptotic behaviour of ${}_2F_1(a,b=a,c; |z|\rightarrow\infty)$?