Asymptotic expansions for the coefficients of extremal quasimodular forms and a conjecture of Kaneko and Koike

TL;DR

This paper derives an asymptotic formula for the Fourier coefficients of extremal quasimodular forms, demonstrating positivity for most coefficients up to certain depths and weights, and partially confirming a conjecture by Kaneko and Koike.

Contribution

It provides the first asymptotic expansion for these coefficients and verifies the positivity conjecture for a wide range of weights and depths.

Findings

Asymptotic formula for Fourier coefficients derived

Most coefficients of depth ≤ 4 are positive

Conjecture confirmed for weights ≤ 200 and depths 1-4

Abstract

Extremal quasimodular forms have been introduced by M.~Kaneko and M.Koike as as quasimodular forms which have maximal possible order of vanishing at $i\infty$. We show an asymptotic formula for the Fourier coefficients of such forms. This formula is then used to show that all but finitely many Fourier coefficients of such forms of depth $\leq4$ are positive, which partially solves a conjecture stated by M.~Kaneko and M.Koike. Numerical experiments based on constructive estimates confirm the conjecture for weights $\leq200$ and depths between $1$ and $4$.