I want to prove the following recursive relation: $$c(c+1)_2F_1(a,b;c;z)=c(c-a+1)_2F_1(a,b+1;c+2;z)+a[c-(c-b)z]_2F_1(a+1,b+1;c+2;z)$$
I tried using both the series $_2F_1(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{n!(c)_n}z^n$ and the integral representation: $_2F_1(a,b;c;z)=\frac{1}{B(b,c-b)}\int_0^1 t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt$ but I did not manage to prove it.