Large Deviations for the d'Arcais Numbers

The d’Arcais polynomials $P_n(z)$ for $n\in \{0,1,\dots \}$ are defined as ${\sum }_{n=0}^{\mathrm{\infty }}{P}_n(z)q^n=\exp (-z\ln ((q;q{)}_{\mathrm{\infty }}))$ where the $q$-Pochhammer symbol is $(q;q{)}_{\mathrm{\infty }}={\prod }_{k=1}^{\mathrm{\infty }}(1-q^k)$ for $\mathrm{\mid }q\mathrm{\mid }<1$. Denoting the coefficients for $n\in \mathbb{N}$ by the formula $P_n(z)={\sum }_{k=1}^{n}A(2,n,k)z^k\mathrm{/}n!$, we prove that $k_n!A(2,n,k_n)\mathrm{/}n!$ satisfies a Bahadur-Rao type large deviation formula in the limit $n\to \mathrm{\infty }$ with $k_n\mathrm{/}n\to \kappa \in [0,1)$ as long as $k_n\to \mathrm{\infty }$. The large deviation rate function is the Legendre-Fenchel transform $g^{\ast }(-\kappa )$ where $g(\kappa )=f^{-1}(\kappa )$ for the function $f:(0,\mathrm{\infty })\to \mathbb{R}$ given by $f(y)=\ln (-\ln ((e^{-y};e^{-y}{)}_{\mathrm{\infty }}))$. We relate this fact to information about the abundancy index.