Write the following "sum" in terms of $\,_2F_1(a,b;c;z)$:
$$ _2 F_1(a,b+1;c+1;z) +\, _2 F_1(a,b-1;c-1;z) -\, _2 F_1(a,b;c;z).$$
Attempt: I played around with identities in here. In particular, to make necessary transformation, I had to take derivative w.r.t. $\,z\,$ at some point after which I would get additional functions which I had to convert again to $\,_2F_1(a,b;c;z).$