\[\left(\frac{W_{0}\left(z\right)}{z}\right)^{a}={\mathrm{e}}^{-aW_{0}\left(z%
\right)}=\sum_{n=0}^{\infty}\frac{a\left(n+a\right)^{n-1}}{n!}\left(-z\right)^%
{n},\]