Abstract
Modular and quasimodular solutions of a specific second order differential equation in the upper-half plane, which originates from a study of supersingular j-invariants in the first author's work with Don Zagier, are given explicitly. Positivity of Fourier coefficients of some of the solutions as well as a characterization of the differential equation are also discussed.
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References
M. Kaneko and D. Zagier, “Supersingular j-invariants, Hypergeometric series, and Atkin's orthogonal polynomials,” AMS/IP Studies in Advanced Mathematics 7 (1998), 97-126.
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Kaneko, M., Koike, M. On Modular Forms Arising from a Differential Equation of Hypergeometric Type. The Ramanujan Journal 7, 145–164 (2003). https://doi.org/10.1023/A:1026291027692
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DOI: https://doi.org/10.1023/A:1026291027692