Taylor coefficients of non-holomorphic Jacobi forms and applications
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In this paper, we prove modularity results of Taylor coefficients of certain non-holomorphic Jacobi forms. It is well known that Taylor coefficients of holomorphic Jacobi forms are quasimodular forms. However, recently there has been a wide interest in Taylor coefficients of non-holomorphic Jacobi forms, for example, arising in combinatorics. In this paper, we show that such coefficients still inherit modular properties. We then work out the precise spaces in which these coefficients lie for two examples.
KeywordsCranks Harmonic Maass forms Jacobi forms Joyce invariants Lowering operator Mock modular forms Moments Ranks
Mathematics Subject Classification11F12 11F20 11F37 11F50 11P82 11P83
The author thanks Karl Mahlburg for many enlightening discussions. Moreover, she thanks Byungchan Kim, Jeremy Lovejoy, and Karl Mahlburg for helpful comments on an earlier version of this paper. Finally, she strongly appreciates the many helpful comments of the anonymous referee. The research of the author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement No. 335220—AQSER. This paper is dedicated to Don Zagier who has been a great inspiration in honor of his 65th birthday.
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