The real and complex hypergeometric series $F(a,b;c;z)=\sum_{n=0}^\infty\frac{a_{(n)} b_{(n)}}{c_{(n)}n!}z^n$ is absolutely convergent when the real part of $a+b-c-1$ is less than $-1$. It has several applications in differential equation and integral perhaps because it’s related to exponential, log and power functions, and its derivative has a similar form.
My questions are:
- What’s the historical motivation of this series? Several uses of the series appear here: Motivating Hypergeometric Series, but I’m wondering when the series began to come about and be investigated.
- Since the series looks like a Taylor expansion, if a function $f(z)=F(a,b;c;z)$ (or it has similar expressions), then do we have $f^{(n)}(0) =\frac{a_{(n)} b_{(n)}}{c_{(n)}}$ (or similar conclusions)?
