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

Peter Grabner*

*Corresponding author for this work

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Extremal quasimodular forms have been introduced by Kaneko and Koike as quasimodular forms which have maximal possible order of vanishing at i∞. 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 ≤4 are positive, which partially solves a conjecture stated by Kaneko and Koike. Numerical experiments based on constructive estimates confirm the conjecture for weights ≤200 and depths between 1 and 4.
Original languageEnglish
Pages (from-to)1021-1041
Number of pages21
JournalThe Ramanujan Journal
Issue number3
Early online date2022
Publication statusPublished - Mar 2022


  • Extremal quasimodular forms
  • Fourier coefficients

ASJC Scopus subject areas

  • Algebra and Number Theory

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