What is the infinite series corresponding to the infinite product below? $$f(\alpha,x)=\prod_{n=1}^\infty \left(1+\frac{x}{\alpha^n}\right)$$
Edit: Martin R told me to speak about what I have gotten up to now about how to expand this, so: up to now I only got really complicated expressions with double sums for example. I think this is related to partitions but I don't know exactly how. The reason for thinking this function is related to partitions is because at $x=1$ it is the generating function at $\frac{1}{\alpha}$ for the partition function, and at $\frac{1}{\alpha}$ it actually converges, as long as $\alpha>1$ (for $\alpha\in\mathbb{R}$
Edit 2: I am looking for a clean expression like a power series with some coefficients I don't yet know. I wanted to know this because I was trying to evaluate something else that is related to partitions looking at it now.