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...
+ V_{\beta k}^{*}V_{\gamma q}
\langle T c_{k}^\dagger(t) \eta_\beta(t)\eta_\gamma(t')c_q(t')\rangle \,,
\end{multline}
We apply Wick theorem to $S^t(t,t')$,
the then
\begin{eqnarray}
&&S^t(t,t´)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k} V_{\gamma q}^{*}
\\
...
-\langle T \eta_\beta (t)c^\dagger_q(t')\rangle \langle T \eta_\gamma(t') c_{k}(t)\rangle\Biggr\}
\\
\nonumber
&& + V_{\beta k}^{*}V_{\gamma q}\Biggr\{\langle T c_{k}^\dagger(t) c_q(t')\rangle \langle \eta_\gamma(t')\eta_\beta(t)\rangle -\langle T c_{k}^\dagger(t)\eta_\gamma(t') \rangle\langle
T c_q(t')\eta_\beta(t)\rangle\Biggr\}\,,
\end{eqnarray}
The Green functions for the Majorana-Majorana, Majorana-Lead, and Lead-lead cases are
\begin{eqnarray}
G^{t}_{kq}(t,t') = -i\langle T c_k(t) c^\dagger_q(t') \rangle, \quad\,\, G^{t,h}_{kq}(t,t') = -i\langle T c^\dagger_k(t) c_q(t') \rangle
\end{eqnarray}
\begin{eqnarray}
G^{t}_{\beta\gamma}(t,t') = -i\langle T \eta_\beta(t) \eta_\gamma(t') \rangle,
\end{eqnarray}
\begin{eqnarray}
G^{t}_{k\beta}(t,t') = -i\langle T c_k(t) \eta_\beta(t') \rangle, \quad\,\, G^{t,h}_{kq}(t,t') = -i\langle T c^\dagger_k(t) \eta_\beta(t') \rangle
\end{eqnarray}
\begin{eqnarray}
G^{t}_{\beta k}(t,t') = -i\langle T \eta_\beta(t) c^\dagger_k(t') \rangle, \quad\,\, G^{t,h}_{kq}(t,t') = -i\langle T \eta_\beta(t) c_k(t') \rangle
\end{eqnarray}
Using the definition for the Green functions we can write down the expression for $S^t(t,t')$
\begin{eqnarray}
&&S^t(t,t´)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k} V_{\gamma q}^{*}
[G^t_{\beta\gamma}(t,t')
G^{h,t}_{qk}(t',t) G^{t,h}_{qk}(t',t) - G^{t}_{\beta q}(t,t')G^{h,t}_{\gamma k}(t',t)]\\ \nonumber
&& + V_{\beta k}^{*}V_{\gamma q}
[G^{h,t}_{kq}(t,t') [G^{t,h}_{kq}(t,t') G{t}_{\gamma\beta}(t',t) -
G^{h,t}_{k\gamma}(t,t')G^{t}_{q G^{t,h}_{k\gamma}(t,t')G^{t}_{q \beta}(t',t)]
\end{eqnarray}
Finally, we employ the following relation to obtain $S^{<(>)}(t,t')$:
\begin{equation}
S(t,t') = A(t,t') B(t',t) \rightarrow S^{>(<)}(t,t') = A^{>(<)}(t,t') B^{<(>)}(t',t)
\end{equation}
Now we compute the lead-lead Green function
$G^{h,t}_{kq}(t,t')=\langle $G^{t,h}_{kq}(t,t')=\langle T c_k^\dagger(t) c_q(t')\rangle$ that appears in the previous expression. We compute its equation-of-motion
\begin{equation}
i\hbar \partial_{t'} G_{kq}^{h,t}(t,t') = \epsilon_q
G_{kq}^{h,t}(t,t') G_{kq}^{t,h}(t,t') + \sum_\beta V_{\beta q}^* G^{h,t}_{k\beta}(t,t')
\end{equation}
Then we get
\begin{equation}
G_{kq}^{h,t}(t,t') G_{kq}^{t,h}(t,t') =
G_{kq}^{h,t}(t,t')\delta_{kq} G_{kq}^{t,h}(t,t')\delta_{kq} + \frac{-1}{\hbar}
\sum_\beta\int \sum_\tau\int dt_1
G^{h,t}_{k\beta}(t,t')V_{\beta G^{t,h}_{k\tau}(t,t')V_{\tau q}^*
g_{q}^{h,t}(t_1,t') g_{q}^{t,h}(t_1,t')
\end{equation}
In a similar way we can obtain the mixed (hole) lead-Majorana Green function
$G_{k\beta}^{h,t}{t,t'}$ $G_{k\beta}^{t,h}{t,t'}$
\begin{equation}
G_{k\beta}^{h,t}(t,t') G_{k\tau}^{t,h}(t,t') = \frac{-1}{\hbar}
\sum_\beta\int \sum_{\theta}\int dt_1
g_{k}^{h,t}(t,t_1) V_{\gamma g_{k}^{t,h}(t,t_1) V_{\theta k}
G^{t}_{\gamma\beta}(t,t') G^{t}_{\theta\tau}(t,t')
\end{equation}
Then, inserting the previous expression into the equation for $G_{qk}^{h,t}(t,t')$, we obtain
\begin{equation}
...
d\omega e^{-i\omega (t-t')} \frac{1}{2\pi}\int_{-\infty}^\infty d\epsilon_1 e^{-i\epsilon_1 (t-t')} \frac{1}{2\pi}\int_{-\infty}^\infty d\epsilon_2 e^{i\epsilon_2 (t-t')}
\\
\Biggr\{
V_{\beta k} V_{\gamma q}^{*} [G^>_{\beta\gamma}(\epsilon_1)
G^{h,<}_{qk}(\epsilon_2) G^{<,h}_{qk}(\epsilon_2) - G^{>}_{\beta
q}(\epsilon_1)G^{h,<}_{\gamma q}(\epsilon_1)G^{<,h}_{\gamma k}(\epsilon_2)]
+ V_{\beta k}^{*}V_{\gamma q} [G^{>,t}_{kq}(\epsilon_1) G^{<}_{\gamma\beta}(\epsilon_2) -
G^{h,>}_{k\gamma}(\epsilon_1)G^{<}_{q G^{>,h}_{k\gamma}(\epsilon_1)G^{<}_{q \beta}(\epsilon_2)]\Biggr\}\,,
\end{multline}
Now we enter the expression for
$G_{kq}^{h,t}(t,t')$ $G_{kq}^{t,h}(t,t')$ in the frequency domain
\begin{equation}
G_{qk}^{h,t}(\omega) =
g_{q}^{h,t}(\omega)\delta_{kq} g_{q}^{t,h}(\omega)\delta_{kq} + \sum_{\tau\theta}
g_{q}^{h,t}(\omega) g_{q}^{t,h}(\omega) V_{\tau q} G^{t}_{\tau\theta}(\omega)V_{\theta k}^*
g_{k}^{h,t}(\omega)\,, g_{k}^{t,h}(\omega)\,,
\end{equation}
\begin{multline}
