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\begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\epsilon) V_{\nu q} g^{>}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k} g^{<,h}_{k}(\omega+\epsilon) V_{\beta k}]\\
&[G_{\beta\nu}^r(\epsilon) &+[G_{\beta\nu}^r(\epsilon) V_{\nu q} g^{>}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\omega+\epsilon) V_{\beta k}]
\\
&[G_{\beta\nu}^>(\epsilon) &+[G_{\beta\nu}^>(\epsilon) V_{\nu q} g^{a}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k} g^{<,h}_{k}(\omega+\epsilon) V_{\beta k}]\\
&[G_{\beta\nu}^>(\epsilon) &+[G_{\beta\nu}^>(\epsilon) V_{\nu q} g^{a}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\omega+\epsilon) V_{\beta k}]
\end{align*}
Inserting the expressions for the self-energies we get
\begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^r(\epsilon+\omega)
\Sigma_{0,\mu\beta}^{<,h}(\epsilon+\omega)][G_{\beta\nu}^r(\epsilon) \Sigma_{0,\mu\beta}^{<,h}(\epsilon+\omega)]+[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)]
\\
&[G_{\beta\nu}^>(\epsilon) &+[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon)
G_{\gamma\mu}^r(\omega+\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)][G_{\beta\nu}^>(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)]+[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)]
\end{align*}
\begin{align*}
&N^>(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{
[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\mu}^r(\epsilon+\omega) \Gamma_{\mu\beta} (1-f_e(\epsilon) f_{h}(\epsilon+\omega)] \\
&\sum_{\lambda\delta}[G_{\beta\nu}^r(\epsilon) &+\sum_{\lambda\delta}[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\lambda}^r(\omega+\epsilon) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\epsilon+\omega)[i\Gamma_{\mu\beta}](1-f_e(\epsilon)(f_{h}(\epsilon+\omega)+f_e(\epsilon+\omega))] \\
&\sum_{\lambda\delta} &+\sum_{\lambda\delta} G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\mu}^r(\omega+\epsilon)(1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))f_h(\epsilon)\\
&\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} &+\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\omega+\epsilon)\Gamma_{\theta\tau} G_{\tau\mu}^a(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_{e}(\epsilon)+1-f_{h}(\epsilon))(f_{e}(\epsilon+\omega)+f_{h}(\epsilon+\omega))] \Biggr\}
\end{align*}
Finally, to obtain $N<(t,t)$ we just change $(1-f)\rightarrow f$, and $f\rightarrow (1-f)$.
\begin{align*}
&N(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{
[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\mu}^r(\epsilon+\omega) \Gamma_{\mu\beta} F_{ee} \\
&+\sum_{\lambda\delta}[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma} G_{\gamma\lambda}^r(\omega+\epsilon) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\epsilon+\omega)[i\Gamma_{\mu\beta}](F_{eh}+F_{ee}) \\
&+\sum_{\lambda\delta} G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\mu}^r(\omega+\epsilon)(F_{eh}+F_{hh})\\
&+\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\omega+\epsilon)\Gamma_{\theta\tau} G_{\tau\mu}^a(\omega+\epsilon)[i\Gamma_{\mu\beta}](F_{ee}+F_{hh}+(F_{eh}+F_{he})) \Biggr\}
\end{align*}
The next term is $M(t,t') = M^>(t,t´) + M^<(t,t´)$ with
\begin{eqnarray}
M^>(t,t´)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k}^{*}V_{\gamma q} [G^{h,>}_{kq}(t,t') G^{<}_{\gamma\beta}(t',t)
