The Gauss hypergeometric function F(a, b, c; z), defined by the Taylor series expansion around z = 0 as F(a, b, c; z) = \(\rm \Sigma_{n=0}^\infty\frac{a(a+1)...(a+n-1)b(b+1)...(b+n-1)}{c(c+1)...(c+n-1)n!}z^n\) satisfies the recursion relation
1
\(\rm \frac{d}{dz}F(a,b,c;z)=\frac{c}{ab}F(a-1, b-1, c-1;z)\)
2
\(\rm \frac{d}{dz}F(a,b,c;z)=\frac{c}{ab}F(a+1, b+1, c+1;z)\)
3
\(\rm \frac{d}{dz}F(a,b,c;z)=\frac{ab}{c}F(a-1, b-1, c-1;z)\)
4
\(\rm \frac{d}{dz}F(a,b,c;z)=\frac{ab}{c}F(a+1, b+1, c+1;z)\)