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Taylor coefficients of non-holomorphic Jacobi forms and applications

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Abstract

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.

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References

  1. Andrews, G.: Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks. Invent. Math. 169, 37–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atkin, A., Garvan, F.: Relations between the ranks and the cranks of partitions. Ramanujan J. 7, 343–366 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atkin, A., Swinnerton-Dyer, H.: Some properties of partitions. Proc. Lond. Math. Soc. 4, 84–106 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bringmann, K.: On the explicit construction of higher deformations of partition statistics. Duke Math. J. 144, 195–233 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bringmann, K., Garvan, F., Mahlburg, K.: Partition statistics and quasiharmonic Maass forms. Int. Math. Res. Not. 2009, 63–97 (2008)

    Article  MATH  Google Scholar 

  6. Bruinier, J., Funke, J.: On two geometric theta lifts. Duke Math. J. 125, 45–90 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dyson, F.: Some guesses in the theory of partitions. Eureka (Cambridge) 8, 10–15 (1944)

    MathSciNet  Google Scholar 

  8. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser, Boston (1985)

    Book  MATH  Google Scholar 

  9. Kaneko, M., Zagier, D.: A Generalized Jacobi Theta Function and Quasimodular Forms. The Moduli Space of Curves (Texel Island, 1994). Progress in Mathematics, vol. 129, pp. 165–172. Birkhauser, Boston (1995)

    MATH  Google Scholar 

  10. Mellit, A., Okada, S.: Joyce invariants for \(K3\) surfaces and mock theta functions. Commun. Number Theory 3, 655–676 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rademacher, H.: Topics in Analytic Number Theory. Die Grundlehren der math. Wiss., Band, vol. 169. Springer, Berlin (1973)

  12. Zwegers, S.: Multivariable Appell functions (preprint)

  13. Zwegers, S.: Mock theta functions, Ph.D. Thesis, Universiteit Utrecht (2002)

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Acknowledgements

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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  1. University of Cologne, Cologne, Nordrhein-Westfalen, Germany

    Kathrin Bringmann

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  1. Kathrin Bringmann
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Correspondence to Kathrin Bringmann.

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Bringmann, K. Taylor coefficients of non-holomorphic Jacobi forms and applications. Res Math Sci 5, 15 (2018). https://doi.org/10.1007/s40687-018-0132-6

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