Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane. These led to asymptotic results for the partition numbers and proven and unproven formulas for the coefficients of the reciprocals of Eisenstein series Ek, especially of weight 4. Berndt et al. finally proved them all. Recently, Bringmann and Kane generalized Petersson’s approach via Poincaré series, to handle the general case. We introduce a third approach. We attach recursively defined polynomials to reciprocals of Eisenstein series. This provides easy access to the signs of the Fourier coefficients of reciprocals of Eisenstein series, sheds some light on reciprocals of Ek of general weight, and provides some upper and lower bounds for their growth.

中文翻译：

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爱森斯坦级数的多项式和倒数

Hardy 和 Ramanujan 引入了圆方法来研究某些亚纯模形式在上复半平面上​​的傅里叶展开。这些导致了分区数的渐近结果以及爱森斯坦级数倒数系数的已证明和未证明的公式乙ķ, 特别是重量为 4. Berndt等。终于证明了这一切。最近，Bringmann 和 Kane 通过 Poincaré 级数推广了 Petersson 的方法来处理一般情况。我们介绍第三种方法。我们将递归定义的多项式附加到爱森斯坦级数的倒数上。这提供了轻松访问爱森斯坦级数倒数的傅立叶系数的符号，揭示了倒数乙ķ一般权重，并为它们的生长提供了一些上限和下限。