Faber polynomials and Poincar\'e series

TL;DR

This paper studies weakly holomorphic modular forms for the full modular group, focusing on Faber polynomials, and uses harmonic Maass forms and Poincaré series to analyze their coefficient growth and related asymptotics.

Contribution

It introduces new asymptotic formulas for coefficients of Faber polynomials and derivatives of Maass-Poincaré series, extending previous results to broader contexts.

Findings

Derived asymptotic growth formulas for Faber polynomial coefficients

Extended asymptotics for derivatives of Maass-Poincaré series

Generalized asymptotics for integrals of the Gauss error function

Abstract

In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished set of generators $f_{m,2-k}$. Such weakly holomorphic modular forms have been classified in terms of finitely many Eisenstein series, the unique weight 12 newform $\Delta$, and certain Faber polynomials in the modular invariant $j(z)$, the Hauptmodul for $\SL_2(\Z)$. We employ the theory of harmonic weak Maass forms and (non-holomorphic) Maass-Poincar\'e series in order to obtain the asymptotic growth of the coefficients of these Faber polynomials. Along the way, we obtain an asymptotic formula for the partial derivatives of the Maass-Poincar\'e series with respect to $y$ as well as extending an asymptotic for the growth of the $\ell$-th repeated integral of the Gauss error function at $x$ to include $\ell\in \R$ and a wider range of $x$.