\[W_{\kappa,\pm\frac{1}{2}n}\left(z\right)=\frac{(-1)^{n}e^{-\frac{1}{2}z}z^{%
\frac{1}{2}n+\frac{1}{2}}}{n!\Gamma\left(\frac{1}{2}-\frac{1}{2}n-\kappa\right%
)}\left(\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(\kappa+\frac{1}{2}-\frac{1}%
{2}n\right)_{k}}}z^{-k}-\sum_{k=0}^{\infty}\frac{{\left(\frac{1}{2}n+\frac{1}{%
2}-\kappa\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+\psi\left(%
\tfrac{1}{2}n+\tfrac{1}{2}-\kappa+k\right)-\psi\left(1+k\right)-\psi\left(n+1+%
k\right)\right)\right),\]