Dirichlet series of Rankin-Cohen brackets

Abstract

Given modular forms f and g of weights k and l, respectively, their Rankin-Cohen bracket [f, g](n)((k, l)) 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. (C) 2010 Elsevier Inc. All rights reserved.