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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?

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Turns out, in this particular case Mathematica makes use of a different identity:

Hypergeometric2F1[b, a, a + 1, z] == (a Beta[z, a, 1 - b])/z^a

This identity is not restricted to any particular parameter values, and in the special case z = 1 reduces to

(a Beta[z, a, 1 - b])/z^a /. z -> 1 // FunctionExpand

(Gamma[1 + a] Gamma[1 - b])/Gamma[1 + a - b]

which explains the output.

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