Combinatorics of Maass–Shimura operators

Maass–Shimura operators on holomorphic modular forms preserve the modularity of modular forms but not holomorphy, whereas the derivative preserves holomorphy but not modularity. Rankin–Cohen brackets are bilinear operators that preserve both and are expressed in terms of the derivatives of modular forms. We give identities relating Maass–Shimura operators and Rankin–Cohen brackets on modular forms and obtain a natural expression of the Rankin–Cohen brackets in terms of Maass–Shimura operators. We also give applications to values of L-functions and Fourier coefficients of modular forms.