Evaluating $\lim\limits_{z\to 1}\;(1-z)^{-\gamma}{}_2F_1\left(1,\gamma;1+\gamma;(1-z)^{-1}\right)$

Question

What is

$$\lim_{z\to1}\,(1-z)^{-\gamma}\;{}_2F_1\left(1,\gamma;1+\gamma;(1-z)^{-1}\right)?$$

Here, ${}_2F_1$ is Gauss's hypergeometric function, and $\gamma\in\mathbb{R}^+$.

Answer 1

Using the Euler transformation, we have

$$\frac1{(1-z)^\gamma}{}_2 F_1\left({{1,\gamma}\atop{1+\gamma}}\middle|\frac1{1-z}\right)=(-z)^{-\gamma}{}_2 F_1\left({{\gamma,\gamma}\atop{1+\gamma}}\middle|\frac1z\right)$$

From here, use the Gaussian hypergeometric theorem:

$$(-1)^{-\gamma}{}_2 F_1\left({{\gamma,\gamma}\atop{1+\gamma}}\middle|1\right)=(-1)^{-\gamma}\frac{\Gamma(1+\gamma)\Gamma(1-\gamma)}{\Gamma(1)^2}=(-1)^{-\gamma}\pi\gamma\,\csc(\pi\gamma)$$