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I need to evaluate this series for arbitrary $\beta > 0$:

$ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $

Is it related to a known transcendental function?

From the research I did, it doesn't seem to be so; in which case I would be interested in knowing of good (piecewise?) approximation of it, one that minimizes discontinuities and is monotonic on $\beta$, or an alternative fast converging series (e.g. one that needs less than $\sim\sqrt{1/\beta}$ terms for small $\beta$)

By the way, this is related to the partition function of the quantum rigid rotor.

Bonus: same question for these subseries:

$ Q_\text{even} = \sum_{J=0, J \text{is even}}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $

$ Q_\text{odd } = \sum_{J=1, J \text{is odd}}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $

References to existing numerically implementations are also welcomed.

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These are related to theta-functions, q.v. – Gerry Myerson Sep 28 '12 at 5:58
Since the integral $\int_0^\infty (2x+1)e^{-\beta x(x+1)}\ dx$ is well convergent and can be explicitly computed you should maybe look at the Euler-Maclaurin summation formula. See here: – Christian Blatter Sep 28 '12 at 20:49

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