Rankin-Cohen operators and their applications

Modular forms have not lost popularity for more than a hundred years and are still widely studied, as they bring together many branches of mathematics and their applications extend to physics. In this study, the subject of Rankin-Cohen operators (brackets), which is located at the intersection of analysis and function theory and modular forms. The first part of this thesis, which consists of 7 chapters and an appendix, is devoted to preliminaries and the introduction of topics that will be used in the thesis. In the second part, differential operators defined on modular forms are given. In the third part, the concept of Rankin-Cohen bracket, which forms the main part of the thesis, is introduced and its basic features are examined. In the fourth section, the relation of Rankin-Cohen bracket with Petersson inner product defined on modular forms is discussed. In the fifth chapter, the article published in 2011 by Choie and Lee is examined. In the sixth chapter, an interesting result about the Rankin-Cohen parenthesis given by Lanphier and Takloo-Bighash is discussed. As an application of the Rankin-Cohen bracket, which forms the seventh and original part of the thesis, the two Hecke eigenforms in the Kohnen-plus space are expressed via the first order Rankin-Cohen parenthesis in terms of the classical theta series and Eisenstein series are expressed. In addition to the thesis, there are screenshots of some calculations including the Rankin-Cohen bracket included in the mf-package in Pari-GP software.