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