Specifically, I'm trying to simplify $_2F_1(1/n,-1/n,1+1/n,z)$.
I see that $_2F_1(a+1,b,a,z)$ and $_2F_1(a,b,a,z)$ are better, but I get a zero coefficient when I try to use the standard contiguous identities for a linear relationship. My original form also appears to be the incomplete Beta function. Is there a sequence of identities I can apply to find a closed form representation?