Let \begin{eqnarray*} P_{k}(z)={_2F_1}(-k,\frac{1}{2}-k; -2k; z), \ \label{e0} Q_{k}(z)={_2F_1}(-k,-\frac{1}{2}-k; -2k; z) , \end{eqnarray*} where $k\ge 1$ is an integer.
How to show \begin{eqnarray*}\label{e1} \frac{Q_{k}(1-z^2)-zP_{k}(1-z^2)}{Q_{k}(1-z^2)+zP_{k}(1-z^2)}=\left(\frac{1-z}{1+z}\right)^{2k+1} \end{eqnarray*} analytically?