Asking for a reference which proves analytic continuation of Hypergeometric Function

Question

I am self studying a research paper in analytic number theory and it requires analyticity of Hypergeometric Function .

I searched google but i couldnot found anything precize on how to prove analytic continuation of this function . Can someone please suggest some reference book or article for it .

Answer 1

A hypergeometric series satisfies (in the region where it converges) a homogeneous linear differential equation with polynomial coefficients. In a neighbourhood of any line segment in the complex plane that avoids the singularities, this will have analytic solutions.