\[\phi_{n}(x;\lambda)={\mathrm{e}}^{-\lambda\,{\mathrm{e}}^{-x}}\left(2\lambda\,%
{\mathrm{e}}^{-x}\right)^{\lambda-\frac{1}{2}}\ifrac{y_{n}\left(\lambda^{-1}{%
\mathrm{e}}^{x};2-2\lambda\right)}{n!}=(-1)^{n}{\mathrm{e}}^{-\lambda\,{%
\mathrm{e}}^{-x}}\left(2\lambda\,{\mathrm{e}}^{-x}\right)^{\lambda-n-\frac{1}{%
2}}L^{(2\lambda-2n-1)}_{n}\left(2\lambda\,{\mathrm{e}}^{-x}\right)=(2\lambda)^%
{-\frac{1}{2}}\,{\mathrm{e}}^{x/2}\,W_{\lambda,n+\frac{1}{2}-\lambda}\left(2%
\lambda{\mathrm{e}}^{-x}\right)/n!\,,\]