How to nicely expand a Gauss Hypergeometric function?

Does anybody know how to obtain the z->1 expansion for the Gauss Hypergeometric 2F1(a,b;c;z) on Mathematica as shown here ? I tried to use Series with the assumption c-a-b non-integer, but the result is given in terms of Floor, Ceiling… I would like to have a result in terms of the Gamma functions, if possible, or other nice functions. I tried following the answer with FunctionExpand and the one with Inactive[Sum] given here
, but with no success.

Cheers