Abstract
We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function \(\wp (z|\tau )\) as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups \(\Gamma _0(N)\) and \(\Gamma _1(N)\).
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Acknowledgements
We thank Prof. Frank Garvan for bringing our attention to the work of Prof. Kolberg and thank Prof. Zhi-Guo Liu and Prof. Li-Chien Shen for their patient guidance. We are also grateful to the referees for their valuable comments.
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The authors were supported in part by the National Natural Science Foundation of China (Grant No. 11971173) and ECNU Short-term Overseas Research Scholarship for Graduate Students (Grant Nos. 201811280046 and 201811280047).
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Chen, D., Chen, R. On a class of elliptic functions associated with even Dirichlet characters. Ramanujan J 55, 175–203 (2021). https://doi.org/10.1007/s11139-020-00292-9
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DOI: https://doi.org/10.1007/s11139-020-00292-9