The hypergemoetric function $_2F_1(a,b,c,z)$ happens to be the primitive of a well known function in astroparticle physics. Result integration by WolframAlpha
In my work, this object is part of more complex function, which is minimized by varying the values of the parameters $a,b,c$.
The question is: for $z<0$, what are the bounds on the parameters $a,b,c$ for $_2F_1(a,b,c,z)$ to be positive definite? Thank you very much for your comments and help.