When calculating a Partition function, I encounter the following summation $$\sum_{n=0}^{\infty} x^n q^{n^2}.$$

I know that the sum $\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function, but I do not know how to perform the sum from 0.

Can anyone help?

By the way I would like to have the result in an infinite product form, which I can easily do in the Theta case using Jacobi's triple product identity.

I want to make a small addition to this question which might or not be related but is also relevant for the physical problem I want to solve. The usual formulas for the infinite product generating functions involve products over 1 integer. To be more concrete one has for example $$\sum_{n=0}^{\infty} \frac{x^n q^{n(n-1)/2}}{(1-q) ... (1-q^n)} = \prod_{m=0}^{\infty} (1+ x q^m).$$ which is the Partition for fermions.

Do you know any reference where people study double products like $$\prod_{m,n=0}^{\infty} (1+ x q^{(n \pm m)}).$$ or maybe $$\prod_{m,n=0}^{\infty} (1+ x q^{(n m)}).$$

Can then one write them as infinite sums with one label as $$\sum_{n=0}^{\infty} x^n A^{n}?$$

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    $\begingroup$ The term "partial theta function" can be found for such sums. Of course they do not have the nice theory of the theta functions. $\endgroup$ – Gerald Edgar Oct 21 '15 at 15:38
  • $\begingroup$ Ok thank you, I am currently trying to understand some of the properties of Partial Theta's and Mock Theta's- do you know any reference with infinite product representations? $\endgroup$ – Panagiotis Betzios Oct 21 '15 at 16:00
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    $\begingroup$ These objects are very different than complete theta functions or mock theta functions. For instance this will not have a nice transfomration fomrula to it. The expansions are not nearly as nice either. There is one paper which studies the estimates for series like this that I know: math.uiuc.edu/~berndt/articles/Asym_PAMS_final.pdf $\endgroup$ – Daniel Parry Oct 21 '15 at 16:09

Those are usually called partial theta functions. They were probably first studied in some depth by Ramanujan, and seem to be closely related to general q-series, mock modular forms. They also show up in combinatorics and statistical physics.

They are not necessarily as well behaved as the complete ones, and usually not modular, but some basic properties such as convergence for $|q|<1$ still holds.

In particular, you still have an infinite product. Using q-Pochhammer notation, I think you can recover what you are looking for from the following Jacobi-type identity:

$$1+\sum_{n=1}^\infty (-1)^nq^{n(n-1)/2}(a^n+b^n)=(q)_\infty(a)_\infty(b)_\infty \sum_{n=1}^\infty \frac{(ab/q)_{2n}q^n}{(q)_n(a)_n(b)_n(ab)_n}$$

This is proved in section 2 of the paper:

See that you can get Jacobi's triple product identity by taking $b=q/a$.

For more general information about partial thetas here are some references:

  • $\begingroup$ Thanks a lot , the Warnaar review is very good. Still probably I cannot get what I want but I will try, an other idea I have is to use a formula by Heine recursively to get what i need.... $\endgroup$ – Panagiotis Betzios Oct 21 '15 at 17:12
  • $\begingroup$ @PanagiotisBetzios Glad to be of some help. $\endgroup$ – Myshkin Oct 21 '15 at 17:57

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