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Ramanujan–Shen’s differential equations for Eisenstein series of level 2

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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), Nidelan (2022) and Toh (2011). As application, we show some arithmetic results on Ramanujan’s tau function.

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All data generated during this study are included in this article. We have no conflicts of interest to disclose.

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Acknowledgements

This work is based on the author’s talk in Hiroshima-Sendai Number Theory Workshop at Hiroshima University, Japan, in July 2023. He thanks its organizers and the audience who gave valuable comments and suggestions. He is also grateful for the anonymous referees for careful reading and helpful comments to improve the manuscript.

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  1. Department of Engineering, Kanagawa University, 3-27-1 Rokkaku-bashi, Yokohama, 221-8686, Japan

    Masato Kobayashi

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  1. Masato Kobayashi
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Kobayashi, M. Ramanujan–Shen’s differential equations for Eisenstein series of level 2. Res. number theory 10, 41 (2024). https://doi.org/10.1007/s40993-024-00527-4

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