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Let $\alpha \in (0,1)$ and $z\geq 0$. Define the function:

$$ \theta_{\alpha}(z)=\sum_{j=0}^{\infty}\left(\frac{z}{\alpha j + 1}\right)^j. $$

Can $\theta_{\alpha}(z)$ be written in closed form? Are there any special functions I could use to represent this function?

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