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.



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