Faber polynomials and Poincaré series

DOI10.4310/MRL.2011.V18.N4.A2zbMATH Open1296.11038arXiv1010.2176MaRDI QIDQ2392980

Publication date: 5 August 2013

Published in: Mathematical Research Letters (Search for Journal in Brave)

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

for the full modular group

. 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

D e l t a

, and certain Faber polynomials in the modular invariant

j (z)

, the Hauptmodul for

. 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

e l l

-th repeated integral of the Gauss error function at

x

to include

e l l i n R

and a wider range of

x

.

Full work available at URL: https://arxiv.org/abs/1010.2176

zbMATH Keywords

Poincaré series special functions Faber polynomials harmonic weak Maass forms Gauss error function

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Forms of half-integer weight; nonholomorphic modular forms (11F37) Holomorphic modular forms of integral weight (11F11) Applications of automorphic functions and forms to multiplicative problems (11N75)