We study quasimodular forms of depth [Formula: see text] and determine under which conditions they occur as solutions of modular differential equations. Furthermore, we study which modular differential equations have quasimodular solutions. We use these results to investigate extremal quasimodular forms as introduced by M. Kaneko and M. Koike further. Especially, we prove a conjecture stated by these authors concerning the divisors of the denominators occurring in their Fourier expansion.

中文翻译：

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作为模微分方程解的拟模形式

我们研究深度的准模形式[公式：见正文]，并确定它们在什么条件下作为模微分方程的解出现。此外，我们研究了哪些模微分方程具有拟模解。我们使用这些结果来研究 M. Kaneko 和 M. Koike 进一步介绍的极值准模形式。特别是，我们证明了这些作者提出的关于分母除数在傅里叶展开中出现的猜想。