G^+_{\Ba}Z^+_{\Ba}-\mathcal G^-_{\Ba}Z^-_{\Ba}),
\label{11}$$ where $$\begin{aligned}
&\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}\equiv
\Tr(e^Se^{-\lambda H^{\pm}}e^Te^{-\nu H^{\pm}}),
\\
&\mathcal G^{\pm}_{\Ba}Z^{\pm}_{\Ba}\equiv
\Tr(e^Se^{-\lambda H^{\pm}}e^T(-1)^{\mathcal N}e^{-\nu H^{\pm}}),
\end{aligned}
\label{12}$$ and $$Z^{\pm}_{\Fa}\,=\,\Tr(e^{-\beta H^{\pm}})\,,\quad
Z^{\pm}_{\Ba}\,=\,\Tr((-1)^{\mathcal N}e^{-\beta H^{\pm}})\,.$$ Moreover, for the partition function $Z$ we obtain the representation: $$Z=\frac12\,(Z^+_{\Fa}+Z^-_{\Fa}+Z^+_{\Ba}-Z^-_{\Ba})\,.$$ In the thermodynamic limit the terms with the subscript $\Ba$ are mutually compensated, therefore, in order to obtain $\mathcal G$  it is enough to calculate $\mathcal G^{\pm}_{\Fa}$.

The considered fermion representation is characterized by the existence of the Fock state $|0\rangle$ common for both operators $H^{+}$ and $H^{-}$, and satisfying the relations $c_k|0\rangle=0$, $k\in\mathcal M$. However, the corresponding coherent states over $|0\rangle$, $$\begin{array}{rcl}
&&\mid\!z\big\rangle\,\equiv\,\exp \Big(\sum\limits^M_{k=1} c^\dagger_k
z_k\Big) \mid\!0\big\rangle\,\equiv\,\exp (c^\dagger z)\mid\!0\big\rangle\,,\\ [0.5cm]
&&\big\langle z^*\!\mid\,\equiv\,\big\langle 0\!\mid \exp
       \Big(\sum\limits^M_{k=1} z^*_k c_k \Big)\,\equiv\,
     \big\langle 0\!\mid\exp (z^* c)\,,
\end{array}$$ are different for $H^+$ and $H^-$. Here the short-hand notations $z^*\equiv(z^*_1,\dots,z^*_M)$ and $z\equiv(z_1,\dots,z_M)$ are used for the sets of independent Grassmann parameters $z_k,z^*_k$, $k\in {\mathcal M}$ (it is appropriate to omit the extra index $\pm$ in $z^*$, $z$). Besides, $\sum_{k=1}^{M} c^{\dagger}_kz_k\equiv c^{\dagger} z$, $\prod_{k=1}^M dz_k\equiv dz$. Let us calculate $\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}$ in  using the representation of the trace in the Grassmann integration formalism [@18]: $$\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}=\int dz\,dz^*\,e^{z^*z}
\langle z^*|e^Se^{-\lambda H^{\pm}}e^Te^{-\nu H^{\pm}}|z\rangle.
\label{13}$$

In order to represent the right-hand side of 