Can we simplify this summation?

Question

Is it possible to simplify the following expression in a form without the sum? $$\sum_{i=0}^n(e^x)^i(e^y)^{i^2}$$ equivalent to $$\sum_{i=0}^na^ib^{i^2}$$ where $a=e^x, b=e^y$ If it's not possible, is it nevertheless possible to do so if we take the limit $n\to\infty$ or if we put some constraints on $x$ and $y$?

I've tried rewriting it as a simpler sum first, but couldn't come up with anything.

Edit: so there is no closed form solution. Does this change when we let $a=1$, so that the summation simply becomes the following? $$\sum_{i=0}^nb^{i^2}$$

Thinking that even the infinite sum $\sum_{i=-\infty}^{\infty} a^i b^{i^2}$ has no known closed form (hence is given a special name theta function), I am skeptical about it. — Sangchul Lee, Jul 30 '17 at 04:38

score 0 · Answer 1 · answered Jul 30 '17 at 11:30

0

In terms of the Jacobi Theta function, we have

$$\vartheta_3\left(\ln\left(\frac a{2\pi i}\right),b\right)=\sum_{n=0}^\infty a^nb^{n^2}$$

for appropriate branches of the natural logarithm. No closed forms are known about these functions in general, though now you've got a name you can append to these sorts of sums where the exponent involves $n^2$, which should allow you to research more specific identities and values more easily.

answered Jul 30 '17 at 11:30

Simply Beautiful Art

74,685

What if we let $a=1$? Do we then obtain a closed form solution? – user56834 Aug 06 '17 at 11:22
@Programmer2134 Sadly no lol. However, as seen in the link, there are a few known values. – Simply Beautiful Art Aug 06 '17 at 11:54

Can we simplify this summation?

1 Answers1