Abstract
A quasimodular form \(\phi \) of depth at most \(m\) corresponds to holomorphic functions \(\phi _0, \phi _1, \ldots , \phi _m\). Given nonnegative integers \(\alpha \) and \(\nu \) with \(\nu \le m\), we introduce a linear differential operator \(\mathcal D_{\phi }^{\alpha , \nu }\) of order \(\nu \) on modular forms whose coefficients are given in terms of derivatives of the functions \(\phi _k\). We then show that Rankin–Cohen brackets of modular forms can be expressed in terms of such operators. As an application, we obtain differential operators associated to certain theta series studied by Dong and Mason.
Similar content being viewed by others
References
Choie, Y., Lee, M.H.: Quasimodular forms, Jacobi-like forms, and pseudodifferential operators. arXiv:1007.4823 (2010)
Cohen, P.B., Manin, Y., Zagier, D.: Automorphic Pseudodifferential Operators. Algebraic Aspects of Nonlinear Systems, Birkhäuser, Boston (1997)
Dong, C., Mason, G.: Transformation laws for theta functions. CRM Proceedings of Lecture Notes, vol. 30, pp. 15–26. American Mathematical Society, Providence (2001)
Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms. Progress in Mathematics, vol. 129. Birkhäuser, Boston (1995)
Lee, M.H.: Differential operators on modular forms associated to theta series. Funct. Approx. Comment. Math. (to appear)
Lee, M.H.: Quasimodular forms and Poincaré series. Acta Arith. 137, 155–169 (2009)
Schoeneberg, B.: Elliptic Modular Functions. Springer-Verlag, Heidelberg (1974)
Zagier, D.: Modular forms and differential operators. Proc. Indian Acad. Sci. Math. Sci. 104, 57–75 (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lee, M.H. Differential operators on modular forms associated to quasimodular forms. Ramanujan J 39, 133–147 (2016). https://doi.org/10.1007/s11139-014-9648-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-014-9648-6