Modular Forms, the Shimura Correspondence, and a Generalization of the Selberg Identity

We adapt an identity of Selberg on the Shimura lift of Hecke eigenforms to the case of Rankin-Cohen brackets of Eisenstein series whose level is a fundamental discriminant. To understand this result, we give an exposition to the theory of modular forms of integral weight as well as the Shimura correspondence between integral and half-integral weight forms.