\[\frac{2\pi\mathrm{i}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(%
\frac{1}{2}-\mu-\kappa\right)}W_{-\kappa,\mu}\left(z\right)={\mathrm{e}}^{-%
\kappa\pi\mathrm{i}}W_{\kappa,\mu}\left({\mathrm{e}}^{\pi\mathrm{i}}z\right)-{%
\mathrm{e}}^{\kappa\pi\mathrm{i}}W_{\kappa,\mu}\left({\mathrm{e}}^{-\pi\mathrm%
{i}}z\right).\]