\[E_{1}\left(z\right)=e^{-z}\left(\frac{c_{0}}{z}+\frac{c_{1}}{z(z+1)}+\frac{2!c%
_{2}}{z(z+1)(z+2)}+\frac{3!c_{3}}{z(z+1)(z+2)(z+3)}+\cdots\right),\]