Infinite Sum of Powers: Is There a Closed Form for the Series?

[stevendaryl]

This isn't quite a calculus question, but it didn't seem right for any of the other mathematics forums, either.

Does anybody if there is a closed form for the following infinite series:

$\sum_n x^{n^2}$

for $0 < x < 1$

 

[mathman]

I strongly doubt it.

 

[Matterwave]

Wolfram alpha gives it in terms of an elliptic theta function...no idea what that is:

http://www.wolframalpha.com/input/?i=1+x+x^4+x^9+...

Looking at the wikipedia page, it looks like these functions are defined through infinite series such as the one seen in the OP...

 

[mfb]

WA does not even have a closed form for x=1/2. Does not look good.

 

[Matterwave]

I just realized, you have to include the ...'s at the end of that URL or else it doesn't work just by clicking on the link.

 

[economicsnerd]

If you could derive one, my best guess of how would be by some clever combinatorial argument, viewing it as a generating function. Just a guess...