What's the asymptotic behavior of ${_2F}_1(-a,n;1-a;z)$ when $a\to 0$?

Question

Consider a function $f(a)={_2F}_1(-a,n;1-a;z)$, where $n\in\mathbb{Z}$, $0\le a<1$, $z<0$, and ${_2F}_1$ is a Gauss hypergeometric function.

Then what's the asymptotic behavior of $f(a)$ when $a\rightarrow 0$?

Answer 1

Using the Pfaff transformation, we have

$$f(a)={}_2 F_1\left({{-a,n}\atop{1-a}}\middle|z\right)=(1-z)^{1-n}{}_2 F_1\left({{1,1-a-n}\atop{1-a}}\middle|z\right)$$

and thus

$$\begin{align*} f(0)&=(1-z)^{1-n}{}_2 F_1\left({{1,1-n}\atop{1}}\middle|z\right)=(1-z)^{1-n}{}_1 F_0\left({{1-n}\atop{}}\middle|z\right)\\ &=(1-z)^{1-n}\left((1-z)^{-(1-n)}\right)=1 \end{align*}$$

where the identical numerator and denominator parameters cancel, and one is left with an elementary hypergeometric series (the binomial series in disguise).