Suppose we know the summation of some series $G(n)$ such that, $$\sum_{n=1}^{\infty}G(n)=S_1.$$ Now assume another summation $S_2$ is expressed as, $$S_2=\sum_{n=1}^\infty G(n) e^{i\frac{2\pi}{m}n}; \text{m is an integer}.$$ Can we evaluate $S_2$ as a function of $S_1$?
To be more specific, and show the steps, the summation $S_1$ is, $$S_1=\sum_{n=1}^\infty \frac{e^{ik\sqrt{a^2+(nb)^2}}}{\sqrt{a^2+(nb)^2}},$$ while its value can be calculated numerically using Ewald summation method. The integrand represents the 3D Helmholtz scalar Green's function, with $k$ is an arbitrary wavenumber. Now, to extend this to the 2D case, I need to evaluate the summation $S_2$. I can also apply Ewald method to the $S_2$, however, my question is, given that I already know the value of $S_1$ (under specific conditions of the problem I have, e.g. $a\to 0$, I can also evaluate this summation analytically) can we get $S_2$. Please notice that, the summation $$\sum_{-\infty}^{\infty'}\frac{e^{ikn}}{n}e^{i\frac{2\pi}{m}n}$$ with the prime means $0$ is not included, has an analytical expression refer to "https://doi.org/10.1088/0305-4470/39/36/009".