\[\mathbf{H}_{0}\left(z\right)=\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{J_{2k+1}%
\left(z\right)}{2k+1}=2\sum_{k=0}^{\infty}(-1)^{k}{J_{k+\frac{1}{2}}}^{2}\left%
(\tfrac{1}{2}z\right),\]