On series expansions of zeros of the deformed exponential function

For $q\in (0,1)$, the deformed exponential function $f(x)={\sum }_{n\ge 1}{x}^n{q}^{n(n-1)\mathrm{/}2}\mathrm{/}n!$ is known to have infinitely many simple and negative zeros $\{x_k(q){\}}_{k\ge 1}$. In this paper, we analyze the series expansions of $-x_k(q)\mathrm{/}k$ and $k\mathrm{/}{x}_k(q)$ in powers of $q$. We prove that the coefficients of these expansions are rational functions of the form $P_n(k)\mathrm{/}{Q}_n(k)$ and ${\widehat{P}}_n(k)\mathrm{/}{Q}_n(k)$, where $Q_n(k)\in \mathbb{Z}[k]$ is explicitly defined and the polynomials $P_n(k),{\widehat{P}}_n(k)\in \mathbb{Z}[k]$ can be computed recursively. We provide explicit formulas for the leading coefficients of $P_n(k)$ and ${\widehat{P}}_n(k)$ and compute the coefficients of these polynomials for $n\le 300$. Numerical verification shows that $P_n(k)$ and ${\widehat{P}}_n(k)$ take non-negative values for all $k\in \mathbb{N}$ and $n\le 300$, offering further evidence in support of conjectures by Alan Sokal.