论文标题
极端准排相形式的系数以及kaneko和koike的渐近膨胀
Asymptotic expansions for the coefficients of extremal quasimodular forms and a conjecture of Kaneko and Koike
论文作者
论文摘要
M.〜Kaneko和M.Koike引入了极端的半模拟形式，作为半模型形式，它们在$ i \ infty $上具有最大可能的消失顺序。我们显示了这种形式的傅立叶系数的渐近公式。然后使用该公式表明，这种形式的深度$ \ leq4 $的所有傅立叶系数几乎有限，这部分解决了M.〜Kaneko和M.Koike所说的猜想。基于建设性估计的数值实验证实了权重$ \ leq200 $且深度在$ 1 $至$ 4 $之间的猜想。
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 $4$.
