How to convert $\sum_{x=0}^n e^{-\beta x^2}$ or $\sum_{x=0}^n e^{-\alpha \beta x^2}$ to a closed form?

Question

How to solve $\sum_{x=0}^n e^{-\beta x^2}$ or $\sum_{x=0}^n e^{-\alpha \beta x^2}$ ??

where $\alpha$ = 1/($n$+1) and $\beta$ is just a variable.

Can I find it?

Answer 1

In general you can't find it on closed form.

Some have closed form for $n=\infty$. $$\sum_{j=0}^\infty e^{-j^2}= \frac12\left(1+\vartheta_3\left(0\;,\;\frac{1}{e} \right) \right)$$ $$\sum_{j=0}^\infty e^{-\beta j^2}= \frac12\left(1+\vartheta_3\left(0\;,\;e^{-\beta} \right) \right)\qquad \beta>0.$$ About the Jacobi theta functions : http://mathworld.wolfram.com/JacobiThetaFunctions.html