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arXiv · 2307.15206

Ramanujan-Shen's differential equations for Eisenstein series of level 2

Masato Kobayashi
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Abstract

Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. Motivated by them, we derive new differential equations for Eisenstein series of level 2 from the second kind of Jacobi theta function. This gives a new characterization of a system of differential equations by Ablowitz-Chakravarty-Hahn (2006), Hahn (2008), Kaneko-Koike (2003), Maier (2011) and Toh (2011). As application, we show some arithmetic results on Ramanujan's tau function.

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Masato Kobayashi. 2023-07-27. Ramanujan-Shen's differential equations for Eisenstein series of level 2. https://arxiv.org/abs/2307.15206

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