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I'm trying to solve a set of equations numerically - mainly because I believe it's not possible to do it analytically. The problem is that they involve expressions of the form

$$\sum_{k=1}^{\infty} \exp(-Ak^2+Bk)$$

where $A$ and $B$ are constants. I was hoping to be able to rewrite this in function of $\textit{Jacobi theta}$ functions, but this seems hard when $k$ does not run to minus infinity as well. Is there some way to rewrite this in terms of theta functions or other functions that have been studied before? This would make it possible to exploit some of their known properties, and possibly speed up the numerical work.

Thanks!

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    Have you ever checked that page? http://mathworld.wolfram.com/JacobiThetaFunctions.html2012-06-03
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    I did check that page, some other references mention the single-sided sum as well, but it seems that the cosine which shows up then is just a rephrasing of the problem.2012-06-04
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    Maybe you can rewrite the function by using Jacobi Triple Product. Did you know that? http://mathworld.wolfram.com/JacobiTripleProduct.html2012-06-04
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    @Mathlover : Again, this seems to involve sums running from -infinity to +infinity, and I don't see a way to rewrite my expression in that way. But maybe what I'm asking is just not possible, or maybe not with the Jacobi theta function or related functions? In any case, I'm a bit surprised, I guessed the expression I wrote down above should be fairly standard and studied before...2012-06-06

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