\[\mathbf{E}_{-n}\left(z\right)=-\mathbf{H}_{-n}\left(z\right)+\frac{(-1)^{n+1}}%
{\pi}\sum_{k=0}^{m_{2}}\frac{\Gamma\left(n\!-\!k\!-\!\tfrac{1}{2}\right)}{%
\Gamma\left(k+\tfrac{3}{2}\right)}(\tfrac{1}{2}z)^{-n+2k+1},\]