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?