I'd like to simplify the following expression
$$\sum_{n=1}^\infty \cos\left(\frac{n\pi x}{L}\right)e^{-a\left(\frac{n\pi}{L}\right)^2}-(-1)^{-bn\left(\frac{n\pi}{L}\right)^2}\cos\left(\frac{n\pi x}{L}\right)c^{-b\left(\frac{n\pi}{L}\right)^2}$$
Are there identities I could use to do this, something along the lines of $$\frac{2}{L}\sum_{n=1}^\infty \frac{L}{n\pi}\sin\left(\frac{n\pi x}{L}\right)=\frac{L-x}{L}$$ and $$\frac{2}{L}\sum_{n=1}^\infty \frac{L}{n\pi}(-1)^n \sin\left(\frac{n\pi x}{L}\right)=\frac{-x}{L}$$
The first one is a Theta function. (The second one may also be, modulo a typo.)
See https://en.wikipedia.org/wiki/Theta_function
In particular, $\vartheta(z; t) =1+2\sum_{n=1}^{\infty}e^{i\pi t n^2}\cos(2\pi n z) $.