On this Wolfram Functions page we can find the following identity:
Hypergeometric2F1[a, b, c, 1] == (Gamma[c] Gamma[c - a - b])/ (Gamma[c - a] Gamma[c - b]) /; Re[c - a - b] > 0
In particular, the condition at the end
Re[c - a - b] > 0 restricts the validity of this identity to somewhat specific values of
a,b,c. Out of curiosity, if we make the following substitution:
Hypergeometric2F1[a, b, c, 1]/.c->1+a
(Gamma[1 + a] Gamma[1 - b])/Gamma[1 + a - b]
then we can see an analogous simplification occurring automatically in Mathematica.
However, with the above parameter choice it is not guaranteed that the constraint
Re[c - a - b] > 0 is satisfied, since
Re[c - a - b]/.c->1+a
Re[1 - b]
and the real part of
b, being completely generic, could possibly be bigger than
1. Is the simplification done by Mathematica still correct? What am I missing?