Large Deviations for the d'Arcais Numbers

Abstract

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

23 pages, 4 figures