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On the other hand we have for $G^>_{\beta\gamma}(\epsilon)$ (accordingly with J.S note)  \begin{eqnarray}  G^>_{\beta\gamma}(\epsilon) = \sum_{k\alpha\delta} \int \frac{d\omega}{2\pi} G^r_{\beta \alpha} [V^*_{k\alpha} g^<_{k}(\omega)V_{k\delta} + V_{k\alpha} g^{h,<}_{k}(\omega) V^*_{k\delta}]G^a_{\delta \gamma}]  \end{eqnarray} We need to compute the following product of Green functions: $G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$  \begin{eqnarray}  &&G^>_{\beta\gamma}(\epsilon) G_{kq}^{h,<}(\omega+\epsilon) = \sum_{k\alpha\delta} \int \frac{d\omega}{2\pi} G^r_{\beta \alpha} [V^*_{k\alpha} g^<_{k}(\omega)V_{k\delta} + V_{k\alpha} g^{h,<}_{k}(\omega) V^*_{k\delta}]G^a_{\delta \gamma}] g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} + \sum_{\beta\gamma} [g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,<}(\omega)\nonumber  \\  &&+g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)+g_{k}^{h,<}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)\,,  \end{eqnarray}