Do formulas for the partial derivatives of the hypergeometric function ${_2F_1}$ exist?
I mean I am interested in $$\frac{\partial}{\partial a}\ {_2F_1}(a,b,c,z)$$$$\frac{\partial}{\partial b}\ {_2F_1}(a,b,c,z)$$ $$\frac{\partial}{\partial c} \ {_2F_1}(a,b,c,z)$$
As @Lucian pointed out the derivatives for the hypergeometric function w.r.t. its parameters are given in terms of the differentiated series expansion $$ \partial_a F(a,b;c;z)=\sum_{k=1}^\infty\frac{(a)_k(b)_k}{(c)_k}(\psi(a+k)-\psi(a))\frac{z^k}{k!}, $$ $$ \partial_b F(a,b;c;z)=\sum_{k=1}^\infty\frac{(a)_k(b)_k}{(c)_k}(\psi(b+k)-\psi(b))\frac{z^k}{k!}, $$ and $$ \partial_c F(a,b;c;z)=\sum_{k=1}^\infty\frac{(a)_k(b)_k}{(c)_k}(\psi(c)-\psi(c+k))\frac{z^k}{k!}. $$ For higher order derivatives see here. Only in special cases do these series have closed-forms in terms of elementary functions. Furthermore, by the ratio test one can confirm that these series converge for $|z|<1$ and for certain values of $a$, $b$, and $c$ convergence also takes place at the endpoints $z={-1,1}$.