# q-binomial-like series with exponentials defining probability distribution

Recently I encountered the series

$$f(d) = \frac{1}{(t;t)_\infty} \sum_{k=0}^\infty \frac{(-1)^k t^{\binom{k}{2}}}{(t;t)_k} e^{-t^{d-k}}$$ where $$(t;t)_n = \prod_{i=1}^n (1-t^i)$$, and $$0 < t < 1$$. Though it's not immediately obvious, it can be checked that $$f(d)$$ is a probability density on $$\mathbb{Z}$$, i.e. $$\sum_{d \in \mathbb{Z}} f(d) = 1$$ and $$f(d) > 0$$--this is the context in which it showed up for me. The series defining $$f(d)$$ furthermore looks a lot like the $$q$$-binomial theorem (see the third formula in the Wikipedia q-binomial theorem section, for instance), but with an extra $$e^{-t^{d-k}}$$ in front of each summand.

My question: has this series, or one like it, been considered before somewhere in the literature? More generally, I'm not used to seeing $$q$$-series which also feature usual (non $$q$$-deformed) exponential functions, and would be interested if these have been studied much.