Rankin-Cohen brackets and Serre derivatives as Poincar\'e series

TL;DR

This paper expresses Serre derivatives and Rankin-Cohen brackets of Eisenstein and Poincaré series using Poincaré averaging, deriving identities for the Ramanujan tau function.

Contribution

It provides explicit formulas for Serre derivatives and Rankin-Cohen brackets in terms of Poincaré series, linking these operators to Poincaré averaging.

Findings

Derived identities for the Ramanujan tau function

Expressed Serre derivatives as Poincaré series

Connected Rankin-Cohen brackets with Poincaré averaging

Abstract

We give expressions for the Serre derivatives of Eisenstein and Poincar\'e series as well as their Rankin-Cohen brackets with arbitrary modular forms in terms of the Poincar\'e averaging construction, and derive several identities for the Ramanujan tau function as applications.