Abstract
Kurokawa and Wakayama (Ramanujan J. 10:23–41, 2005) studied a family of elliptic functions defined by certain q-series. This family, in particular, contains the Weierstrass ℘-function. In this paper, we prove that elliptic functions in this family satisfy certain non-linear algebraic differential equations whose coefficients are essentially given by rational functions of the first few Eisenstein series of the modular group.
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References
Chapman, R.: Private communication (2006)
Kurokawa, N., Wakayama, M.: Certain family of elliptic functions defined by q-series. Ramanujan J. 10, 23–41 (2005)
Lang, S.: Elliptic Functions. Addison Wesley, Reading (1970)
Acknowledgements
The authors would like to express their thanks to the referee for his/her very helpful comments and detailed check of computations.
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M. Wakayama’s work is in part supported by Grant-in-Aid for Scientific Research (B) No. 21340011.
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Wakayama, M., Yamamoto, K. Non-linear algebraic differential equations satisfied by a certain family of elliptic functions. Ramanujan J 30, 173–186 (2013). https://doi.org/10.1007/s11139-012-9425-3
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DOI: https://doi.org/10.1007/s11139-012-9425-3