Vol. 297, No. 2, 2018

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Rankin–Cohen brackets and identities among eigenforms

Arvind Kumar and Jaban Meher

Vol. 297 (2018), No. 2, 381–403
Abstract

We investigate the cases for which the Rankin–Cohen brackets of two quasimodular eigenforms give rise to eigenforms. More precisely, we characterise all the cases in a subspace of the space of quasimodular forms for which Rankin–Cohen brackets of two quasimodular eigenforms are again eigenforms. In the process, we obtain some new polynomial identities among quasimodular eigenforms. To prove the results on quasimodular forms, we prove several results in the theory of nearly holomorphic modular forms. These new results in the theory of nearly holomorphic modular forms are of independent interest.

Keywords
eigenforms, nearly holomorphic modular forms, quasimodular forms, Rankin–Cohen brackets.
Mathematical Subject Classification 2010
Primary: 11F25, 11F37
Secondary: 11F30, 11F67
Milestones
Received: 17 September 2017
Revised: 25 February 2018
Accepted: 28 April 2018
Published: 20 December 2018
Authors
Arvind Kumar
School of Mathematical Sciences
National Institute of Science Education and Research, HBNI
Jatni
Khurda
India
Jaban Meher
School of Mathematical Sciences
National Institute of Science Education and Research, HBNI
Jatni
Khurda
India