Extremal quasimodular forms have been introduced by Kaneko and Koike 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 \(\le \,4\) are positive, which partially solves a conjecture stated by Kaneko and Koike. Numerical experiments based on constructive estimates confirm the conjecture for weights \(\le \,200\) and depths between 1 and 4.

中文翻译：

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极值拟模形式的系数的渐近展开以及Kaneko和Koike的猜想

Kaneko和Koike引入了极准拟模形式，作为准模形式，它在\（i \ infty \）处具有最大的消失序。我们展示了这种形式的傅立叶系数的渐近公式。然后使用该公式表明，这种深度\（\ le \，4 \）形式的傅里叶系数（除有限外）全部为正，部分解决了Kaneko和Koike提出的猜想。基于构造性估计的数值实验证实了权重\（\ le \，200 \）和深度在1-4之间的猜想。