I want to calculate the summations of the form $$ S_N^{(0)}(\varepsilon)=\frac{1}{N}\sum_{m=0}^{N}\sqrt{1+\left(1-\varepsilon^2\right)\cos\frac{m\pi}{N}} $$ and $$ S_N^{(1)}(\varepsilon)=\frac{1}{N}\sum_{m=0}^{N}\sqrt{1+\left(1-\varepsilon^2\right)\cos\frac{2m\pi}{2N+1}} $$ where $N$ is an integer, and $\varepsilon\ll1$ is a real parameter. For large enough $N\to\infty$, the above summations can be expressed through the elliptic integral of the second kind. But currently I just want to know whether or not the closed forms exist, or at least the asymptotic behavior of them at $\varepsilon\ll1$ limit.
I would also be interested in some relevant papers or links. Thank you very much.