as the quantum Casimir force for phonons. Expanding $\epsilon_N(|p_z|,\sigma,n)$ in terms of $p=\sqrt{p_z^2+\frac{\hbar^2\beta_{\sigma,n}^2}{R^2}}$ and $\epsilon_D(|p_z|,\sigma,n)$ in terms of $p'=\sqrt{p_z^2+\frac{\hbar^2\alpha_{\sigma,n}^2}{R^2}}$, we have $$\begin{aligned}
\label{eqn:68}
{\mathcal E}_0(R,\bar{n})&=&\frac{1}{2}\sum_{p_z,\sigma,n}\bigg(v(\bar{n})p\left[1-\frac{p}{\sqrt{4Mg\bar{n}}}+\frac{p^2}{8Mg\bar{n}} \right.\nonumber\\
&-& \left. \frac{p^4}{128M^2g^2\bar{n}^2}+...\right]-g\bar{n}\bigg)\nonumber\\&&+\frac{1}{2}\sum_{p_z,\sigma,n}\bigg(v(\bar{n})p'\left[1-\frac{p'}{\sqrt{4Mg\bar{n}}}+\frac{p'^2}{8Mg\bar{n}} \right.\nonumber\\
&-& \left. \frac{p'^4}{128M^2g^2\bar{n}^2}+...\right]-g\bar{n}\bigg).\end{aligned}$$ Here the first and the second summations correspond to the TE and TM modes respectively. Now, replacing the sum over $p_z$ by an integration over $p_z$, we can rewrite Eq.  as $$\begin{aligned}
\label{eqn:69}
{\mathcal E}_0(R,\bar{n})&=&L\bigg(v(\bar{n})\bigg[\hbar (N^{(2)}+D^{(2)})-\frac{\hbar^{2}(N^{(3)}+D^{(3)})}{\sqrt{4Mg\bar{n}}}\nonumber\\
&&+\frac{\hbar^{3}(N^{(4)}+D^{(4)})}{8Mg\bar{n}}-\frac{\hbar^{5}(N^{(6)}+D^{(6)})}{128M^2g^2\bar{n}^2}+...\bigg]\nonumber\\
&&-g\bar{n}(N^{(1)}+D^{(1)})\bigg).\end{aligned}$$ where $q=p/\hbar$, $N^{(-2s)}=\frac{1}{2\pi}\sum_{\sigma,n}\int_{\frac{\beta_{\sigma,n}}{R}}^\infty q^{-2s}(q^2-\beta_{\sigma,n}^2/R^2)^{-d}= \frac{1}{4\sqrt{\pi}}\sum_{\sigma,n}\left(\frac{\beta^2_{\sigma,n}}{R^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}$, $q'=p'/\hbar$, $D^{(-2s)}=\frac{1}{2\pi}\sum_{\sigma,n}\int_{\frac{\alpha_{\sigma,n}}{R}}^\infty q'^{-2s}(q'^2-\alpha_{\sigma,n}^2/R^2)^{-d}= \frac{1}{4\sqrt{\pi}}\sum_{\sigma,n}\left(\frac{\beta^2_{\sigma,n}}{R^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}$, $d\rightarrow1/2$, and $s\rightarrow-1/2,-1,-3/2,-2,-5/2,...$. We already have mentioned various regularization method for $N^{(-2s)}$ and $D^{(-2s)}$ in Sec. \[Sec2\]. Among such integrals cum lattice summations, $N^{(2)}$ and $D^{(2)}$ can be exactly obtained within zeta regularization scheme as we have already mentioned: $N^{(2)}=-0.01417613719/R^2$ and $D^{(2)}=0.000614794033/R^2$ [@Gosdzinsky]. It is absolutely not easy to obtain regularized values of $N^{(-2s)}$ and $N^{(-2s)}$ for $s\le0$ generalizing the works of the authors of Refs. [@Gosdzinsky] and [@Milton1999]. At the end of Sec. \[Sec2\], we have argued that the Casimir force per cross-sectional area of circular-cylinder would be close to that of a square-cylinder as both are topologically same, and as the later specially is having 4-fold symmetry. Evaluation of $N^{(-2s)}$ is reasonably easy for square-cylinder within dimensional regularization 