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arXiv · 2007.13569

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

Peter J. Grabner
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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 $4$.

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Peter J. Grabner. 2020-11-13. Asymptotic expansions for the coefficients of extremal quasimodular forms and a conjecture of Kaneko and Koike. https://doi.org/10.1007/s11139-020-00368-6

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