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