\[\mathsf{k}_{n}\left(2z\right)=\frac{1}{\pi}n!z^{n+1}\sum_{k=0}^{n}(-1)^{k}%
\frac{2n-2k+1}{k!(2n-k+1)!}{\mathsf{k}_{n-k}}^{2}\left(z\right).\]