On Rankin-Cohen brackets of Hecke eigenforms and modular forms of half-integral weight

We generalize the linear relation formula between the square of normalized Hecke eigenforms of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and normalized Hecke eigenforms of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 k"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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.