On Rankin-Cohen Brackets of Hecke Eigenforms and Modular Forms of Half-Integral Weight

TL;DR

This paper extends the understanding of Rankin-Cohen brackets of Hecke eigenforms, providing new formulas relating their inner products and generalizing previous relations for modular forms of half-integral weight.

Contribution

It introduces generalized formulas for Rankin-Cohen brackets of Hecke eigenforms and extends Zagier's Petersson inner product formula to broader cases.

Findings

Generalized linear relations between eigenforms of different weights.

Extended Zagier's formula to Rankin-Cohen brackets involving Eisenstein series.

Enhanced understanding of the structure of modular forms of half-integral weight.

Abstract

We generalize the linear relation formula between the square of normalized Hecke eigenforms of weight $k$ and normalized Hecke eigenforms of weight $2k$, to Rankin-Cohen brackets of general degree. As an ingredient of the proof, we also generalize a formula of Zagier on the Petersson inner product of Rankin-Cohen brackets involving Eisenstein series.