Differential operators on modular forms associated to quasimodular forms

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.