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I am trying to compute some quantity like a partition function for a harmonic oscillator type system and at the end of the computation I am coming across a sum that looks like,

\begin{align} S=\sum_{n=0}^{\infty}e^{an +bn^2} \end{align}

with $a,b \in \mathbb{R}$. How do I find the sum? Any help regarding the same is highly appreciated.

Fermion
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    Essentially a duplicate of https://math.stackexchange.com/questions/153027/rewriting-infinite-sum-as-theta-function?noredirect=1 . Unfortunately there was no answer there. – Robert Israel Mar 14 '24 at 16:59
  • Are you asking for a closed-form formula (which might not exist), a numerical approximation method, or what? – Robert Israel Mar 14 '24 at 18:01
  • @RobertIsrael A Closed-form formula if possible would be best, otherwise, a numerical approximation method will do. – Fermion Mar 15 '24 at 06:01

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