Asymptotic expansions for the coefficients of extremal quasimodular forms and a conjecture of Kaneko and Koike
| Autor: | Peter J. Grabner |
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| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Conjecture Mathematics - Number Theory Mathematics::Number Theory 010102 general mathematics Order (ring theory) 0102 computer and information sciences 01 natural sciences Constructive symbols.namesake Number theory 010201 computation theory & mathematics Fourier analysis FOS: Mathematics symbols Asymptotic formula Number Theory (math.NT) 0101 mathematics Fourier series Mathematics |
| Zdroj: | The Ramanujan Journal. 57:1021-1041 |
| ISSN: | 1572-9303 1382-4090 |
| DOI: | 10.1007/s11139-020-00368-6 |
| Popis: | 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 $4$. Comment: arXiv admin note: text overlap with arXiv:2002.02736 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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