I am currently reading Weil's book : "Elliptic Functions According to Eisenstein and Kronecker" and in page 56 he uses
the well-known transformation formula for theta series $$\sum\limits_{\mu} exp[-t(x+\mu)^2 - 2\pi i\mu y](x+\mu)^a = i^a e^{xy} \left( \frac{\pi}{t} \right)^{a+\frac{1}{2}} \sum\limits_{\nu} exp[-\frac{\pi^2}{t}(y+\nu)^2 - 2\pi i\nu x](y+\nu)^a $$
where $\mu,\nu \in \mathbb{Z}$, $a=0,1$ and $x,y$ reals. I know this has to do with Poisson summation but i am a bit strugling to see it. Any help?
If $a=0$ then remove its terms. If $H$ is the Fourier transform of $h$ then for $x$ real we have the Fourier series $$\sum_n h(x+n)=\sum_k H(k)e^{2i\pi kx}$$ Here $h(x)=e^{-tx^2}$ and $t> 0$ is real, then the result can be extended to $\Re(t) >0,x\in \Bbb{C}$ by analytic continuation.
Completing the square you can plug your $2i\pi y\mu $ term inside $(x+n)^2$.