Abstract
Given a fixed Jacobi cusp form, we consider a family of linear maps between the spaces of Jacobi cusp forms using the Rankin–Cohen brackets, and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Jacobi cusp forms. This is a generalization of the work due to Kohnen (Math Z, 207:657–660, 1991) and Herrero (Ramanujan J, 10.1007/s11139-013-9536-5, 2014) in case of elliptic modular forms to the case of Jacobi cusp forms which is also considered earlier by Sakata (Proc Japan Acad Ser A, Math Sci 74, 1998) for a special case.
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Acknowledgments
The authors would like to thank Sebastián D. Herrero for providing a copy of his paper [8]. The authors would like to thank B. Ramakrishnan for his helpful comments. Finally, the authors thank the referee for their careful reading of the paper and for many helpful comments.
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The first author would like to thank the Council of Scientific and Industrial Research (CSIR), India for financial support. The second author is partially funded by SERB Grant SR/FTP/MS-053/2012.
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Jha, A.K., Sahu, B. Rankin–Cohen brackets on Jacobi forms and the adjoint of some linear maps. Ramanujan J 39, 533–544 (2016). https://doi.org/10.1007/s11139-015-9683-y
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DOI: https://doi.org/10.1007/s11139-015-9683-y