}+\frac{i%
}{2}(\mu -\nu )$ and $y=-\frac{l\hat{y}_{1}}{2\pi
}-\frac{1}{2}(\mu +\nu )$. The denominator of $P$ is $2A\int
d^{2}\mu d^{2}\nu f_{1}(\mu )f_{2}(\nu )e^{\pi i\frac{\mu \nu }{A}
}\sum_{\delta_1,\delta_2 =0}^{1}e^{\pi
i\delta_1\delta_2}\bar{\theta}_{\delta_1}^{'}\theta
_{\delta_2}^{'}$, here $\theta^{\prime} _{\delta }=\theta \left[
\begin{array}{c}
\frac{\delta }{2} \\
0%
\end{array}%
\right] (2x,4\tau )$, $\bar{\theta}^{\prime} _{\delta }=\theta
\left[
\begin{array}{c}
\frac{\delta }{2} \\
0%
\end{array}%
\right] (2y,4\tau )$. This formula gives another result compared with the Boca’s when $f_{1}(z)=f_{2}(z)=\delta ^{2}(z-0).$

Finally, we will give another explicit form of $P$ in terms of the derivative of elliptic functions. Note that the basis $\{|n>\}$ of Fock space produce a phase $\omega ^{n}$ under action of $R_{N}$. It is not difficult to find that $e^{i\alpha _{1}}$ and $e^{i\alpha _{2}}$ in ([eq:50]{}) are both integral powers of $\omega _{N}$ because of $%
(R_{N})^{N}=identity.$ Therefore $|\phi _{1}>$ and $<\phi _{2}|$ can respectively be expanded in the basis $\{|n>\}$ and $\{<n|\}$, where$$|n>=\frac{(a^{+})^{n}}{\sqrt{n!}}|0>.$$We have the relation between coherent state and particle number eigenstate as following:$$<z|n>=e^{-\frac{1}{2}z\bar{z}}\frac{z^{n}}{\sqrt{n!}}.
\label{eq:43}$$

Obviously, the general forms of $|\phi _{1}>$ and $<\phi _{2}|$ in the expansion in terms of particle number eigenstates are $\sum_{m=0}^{\infty }c_{m}$ $|i+4m>$ and $\sum_{n=0}^{\infty
}d_{n}$ $<j+4n|,$ where $i$ and $j$ are nonnegative integers and $c_{m}$ and $d_{n}$ are arbitrary constant coefficients. So$$\begin{aligned}
|\phi _{1} 