\[\mathbf{L}_{\nu-1}\left(z\right)-\mathbf{L}_{\nu+1}\left(z\right)=\frac{2\nu}{%
z}\mathbf{L}_{\nu}\left(z\right)+\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma%
\left(\nu+\tfrac{3}{2}\right)},\]