Ramanujan-Shen's differential equations for Eisenstein series of level 2
Number Theory
2024-04-16 v2
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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Cite
@article{arxiv.2307.15206,
title = {Ramanujan-Shen's differential equations for Eisenstein series of level 2},
author = {Masato Kobayashi},
journal= {arXiv preprint arXiv:2307.15206},
year = {2024}
}
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21 pages