-s^{\prime }-\nu }{A}%
)u}.\end{aligned}$$where $$C_{1}=e^{2\pi i[\frac{-i}{4A}(\mu ^{2}+\nu ^{2})+\frac{i}{4A}(|\mu
|^{2}+|\nu |^{2})]}.$$It can be proved that for integer $A$: $$\begin{aligned}
&&\sum_{r=0}^{A-1}e^{2\pi irt/A}\theta (x+r/A,\tau /A)\theta
(y+r/A,\tau /A)
\nonumber \\
&=&A\sum_{d=0,1}\theta (-\frac{\tau }{A}(Ad-t)+x-y,\frac{2\tau }{A})\theta (%
\frac{\tau }{A}(-At+A^{2}d)+A(x+y),2\tau A)  \nonumber \\
&&\times e^{\pi i\frac{\tau }{A}(Ad-t)^{2}}\times e^{2\pi
i(Ad-t)y}\end{aligned}$$and $$\theta (z,\tau )=\sqrt{\frac{i}{\tau }}e^{-\pi iz^{2}/\tau }\theta
(\pm \frac{z}{\tau },-\frac{1}{\tau }).$$Thus we have $$\begin{aligned}
  \label{eq:32}
G_{st}(u,v) &\equiv &g_{st}(k,q,z_{1},z_{2})  \nonumber \\
&=&e^{2\pi ivt}\sum_{r}e^{2\pi irt/A}K_{s+r,r}(u,v)  \nonumber \\
&=&\frac{AC_{1}}{l\sqrt{\pi }}\sqrt{\frac{A}{2}}\sum_{d=0,1}e^{-\frac{\pi }{%
2A}(Ad-t)^{2}+2\pi i(Ad-t)\frac{\nu }{A}-\frac{\pi }{2A}(s+\mu
-\nu )^{2}}
\nonumber \\
&&\times e^{\frac{\pi i}{A}(s+\mu -\nu )(Ad-t)+2\pi ivAd}\theta (u-\frac{1}{2%
}(Ad-t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2})  \nonumber \\
&&\times \theta (2Av+s+\mu +\nu -t\tau +iAd,2Ai).\end{aligned}$$Due to $$\begin{aligned}
\sum_{a=0,1}\theta (x+\frac{a}{2},\tau ) &=&2\theta (2x,4\tau ), \\
\sum_{a=0,1}(-1)^{a}\theta (x+\frac{a}{2},\tau ) &=&2e^{2\pi
i(x+\frac{\tau }{2})}\theta (2x+2\tau ,4\tau ),\end{aligned}$$when $d$ is equal to $0,1$, $$\theta (2x+2\tau d,4\tau )=\frac{1}{2}\sum_{a=0,1}(-1)^{ad}e^{2\pi i(x+\frac{%
\tau }{2})d}\theta (x+\frac{a}{2},\tau ).$$The function $G_{st}(u,v)$ can be rewritten as $$\begin{aligned}
G_{st}(u,v) &=&\frac{A}{4\pi }e^{2\pi i\phi
}\sum_{a,d=0,1}(-1)^{ad}\theta
(-u+\frac{1}{2}(Ad+t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2})
\nonumber
\label{eq:13} \\
&&\times \theta (-Av+\frac{1}{2}(s+a-\mu -\nu
)+\frac{i}{2}t,\frac{Ai}{2}),\end{aligned}$$where $$\phi =\frac{i}{4A}(s^{2}+t^{2})-\frac{st}{2A}+\frac{i}{2A}s(\mu -\nu )-\frac{%
t}{2A}(\mu +\nu )+\frac{i}{4A}(|\mu |^{2}+|\nu |^{2}-2\mu \nu )$$From the above discussion, we know that when $A$ is an even number, only $%
a=0 $ contributes, so we have $$G_{st}(u,v)=\frac{A}{2\pi }e^{2\pi i\phi}\theta (-u+\frac{1}{2}t+\frac{i%
}{2}(s+\mu -\nu ),\frac{Ai}{2})\theta (-Av+\frac{1}{2}(s-\mu -\nu )+\frac{i}{2}t,\frac{Ai%
}{2})$$and when $A$ is an odd number, $$\begin{aligned}
G_{st}(u,v) 