can we find a closed form for following sum:
\begin{equation} \sum_{i=0}^{n} a^{i^2} = a^0 + a^1 + a^4+a^9 +...a^{n^2} \end{equation}
where $0<a<1$ , what if $n \rightarrow \infty$ ?
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http://mathworld.wolfram.com/JacobiThetaFunctions.html – Jack D'Aurizio Nov 03 '16 at 22:37
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http://math.stackexchange.com/questions/1974533/find-the-generating-function-for-displaystyle-sum-k-0-infty-xk2 – Nov 03 '16 at 23:00
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Computing the truncated theta function via Mordell integral : https://arxiv.org/pdf/1306.4081v2.pdf – Nov 03 '16 at 23:06
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A NEARLY-OPTIMAL METHOD TO COMPUTE THE TRUNCATED THETA FUNCTION, ITS DERIVATIVES, AND INTEGRALS : https://www.dtc.umn.edu/publications/reports/2008_06.pdf – Nov 03 '16 at 23:07
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@JackD'Aurizio thank you, so it is a Truncated Theta function, do you know any famous approximations or bound for the theta function? – Alireza Nov 04 '16 at 01:48
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@arthur thank you, I skimmed the papers you shared, it seems authors proposed numerical methods for fast evaluation of truncated theta functions, however I am looking for a approximation/bound/etc that can be written in closed form – Alireza Nov 04 '16 at 01:56