$E_m\left(z\right)$

$=:{\left(z+E\left(0\right)\right)}^m$

$E_{m+1}\left(z\right)$

$=:z{E}_m\left(z\right)-\frac{1}{2}{E}_m\left(E\left(z+1\right)\right)$

$E_{m+1}\left(z+y-1\right)$

$=\left(z+y-1\right)E_m\left(z+y\right)-\frac{1}{2}{\left(E\left(y\right)+E\left(z\right)\right)}^m$

$E_{m+1}\left(z\right)$

$=:\left(z-1\right)E_m\left(z\right)+\frac{1}{2}{\left(E\left(0\right)+E\left(z\right)\right)}^m$

$E_{m+1}\left(0\right)+E_m\left(0\right)$

$=:\frac{1}{2}{\left(E\left(0\right)+E\left(0\right)\right)}^m$