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 + 2 n , 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.