Dirichlet series of Rankin–Cohen brackets

Given modular forms f and g of weights k and ℓ, respectively, their Rankin–Cohen bracket corresponding to a nonnegative integer n is a modular form of weight k+ℓ+2n, and it is given as a linear combination of the products of the form f(r)g(n−r) for 0⩽r⩽n. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin–Cohen brackets.