Bose-Fermi mixture and <PERSON> integrals
The problem
Suppose we have a mixture of $N$ fermionic and $M$ bosonic degrees of freedom. The fermionic degrees of freedom are described by $N$ <PERSON> variables $ψ_i$, $i = 1 . . . N$ and the bosonic degrees are described by $M$ complex variables $φ_i$, $i = 1 . . . 346
M$. We combine these into a single vector
$$ X=\begin{pmatrix}\Psi\ \Phi\end{pmatrix}\quad\text{with}\quad\Psi=\begin{pmatrix}\psi_1\ \vdots \ \psi_N\end{pmatrix}\quad\text{and}\quad\Phi=\begin{pmatrix}\phi_1\ \vdots \ \phi_M\end{pmatrix} $$
We also define a matrix $G$ of both complex numbers and <PERSON> numbers as
$$ G=\begin{pmatrix} A & B\ C&D \end{pmatrix} $$
where $A$ and $D$ are resp. $N\times N$ and $M\times M$ matrices which consist of normal complex numbers. $B$ and $C$ are resp. $N\times M$ and $M\times N$ matrices of <PERSON> numbers.
Now we define 'action' $S(X^\dagger,X)=X^\dagger G X$ and $$ Z[G]=\int\mathcal{D}X^\dagger\mathcal{D}X \exp[S(X^\dagger,X)]\quad\text{with}\quad\mathcal{D}X^\dagger\mathcal{D}X=\prod_{i=1}^N d\psi_i^d\psi_i\prod_{j=1}^M\frac{d\phi_j^d\phi_j}{2\pi i} $$ which I want to compute.
Attempt at solution
For simplicity I first want to compute $Z[G]$ in the case $N=M=1$, i.e. there is only one fermionic degree and one bosonic degree of freedom. I write
$$ G=\begin{pmatrix} \alpha & \beta\ \gamma & \delta \end{pmatrix}\quad\text{and}\quad X=\begin{pmatrix}\psi\ \phi\end{pmatrix} $$
The action now reduces to
$$S(X^\dagger,X)=\alpha\psi^\psi+\delta\phi^\phi+\psi^\beta\phi+\phi^\gamma\psi$$
and the partition function $Z[G]$ becomes
$$ \begin{align} Z[G]&=\int d\psi^d\psi\int\frac{d\phi^d\phi}{2\pi i}e^{-\alpha\psi^\psi}e^{-\delta\phi^\phi}e^{-\psi^\beta\phi}e^{-\phi^\gamma\psi}\ &=\int d\psi^d\psi\int\frac{d\phi^d\phi}{2\pi i}(1-\alpha\psi^\psi)(1-\psi^\beta\phi)(1-\phi^\gamma\psi)e^{-\delta\phi^\phi}\ &=\int d\psi^d\psi\int\frac{d\phi^d\phi}{2\pi i}(\beta\gamma\psi^\psi\phi^\phi-\alpha\phi^\phi)e^{-\delta\phi^\phi} \end{align} $$
where I used that $\psi^2=(\psi^)^2=0$, the anti-commuting properties of $\beta,\gamma,\psi,\psi^$ and that integrating over a single $\psi$ or $\psi^*$ yields zero. 364