I need to evaluate this series for arbitrary $\beta > 0$:
$ Q = \sum\limits_{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\limits_{J=0, J \text{is even}}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $
$ Q_\text{odd } = \sum\limits_{J=1, J \text{is odd}}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $
References to existing numerically implementations are also welcomed.