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Consider the following polynomial series:

$S(x) = \sum_{i=1}^{\infty}(-1)^{i+1}x^{i^{2}}$

Between 0 and 1, this looks like a well-behaved function - is there any way to write this function in this interval without using a series?

Given $0 < S(x) < 1$, I need to solve the equation for $x$ (in the 0 to 1 interval), but an analytic solution would be much nicer than a numerical one...

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You can turn that series into a theta function with little effort: – Dan Piponi Apr 23 '10 at 21:42
up vote 4 down vote accepted

I don't know what you mean by "polynomial series" since your function doesn't seem to have much to do with polynomials (perhaps you could elaborate?).

$S(x) = -\frac12 \theta(\frac12, \frac{\log x}{\pi i }) - 1$, where $\theta$ is Jacobi's theta function. I don't know of any nice algebraic methods to take inverses of modular forms (even if you restrict to real $x$, i.e., $\tau$ purely imaginary), so I'm not sure if this is the sort of answer you want.

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Of course, in mathematics (as in any other aspect of life, really) one can't always get the sort of answers one wants. – Harald Hanche-Olsen Apr 24 '10 at 0:41

As other correspondents have pointed out, this is essentially a theta function. You ask if you can write it in any other way. You can replace the infinite series by an infinite product :-) One gets $$1-2S(x)=\prod_{n=1}^\infty(1-x^{2m-1})^2(1-x^{2m}).$$ The equivalence is a special case of the Jacobi triple product: . Using this represention makes numerical computation of the function more difficult :-)

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