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On Fourier–Jacobi expansions of real analytic Eisenstein series of degree 2

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Abstract

We discuss the Fourier–Jacobi expansion of certain vector valued Eisenstein series of degree \(2\), which is also real analytic. We show that its coefficients of index \(\pm 1\) can be described by using a generating series of real analytic Jacobi forms. We also describe all the coefficients of general indices in suitable manners. Our method can be applied to study another Fourier series of Saito-Kurokawa type that is associated with a cusp form of one variable and half-integral weight. Then, following the arguments in the holomorphic case, we find that the Fourier series indeed defines a real analytic Siegel modular form of degree 2.

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Acknowledgments

The author express his sincere gratitude to Professor Shun Shimomura for teaching him how to estimate the Humbert’s confluent hypergeometric function discussed in Sect. 5 of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Keio University, Hiyoshi, Yokohama, 223-8522, Japan

    Takuya Miyazaki

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  1. Takuya Miyazaki
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Correspondence to Takuya Miyazaki.

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Communicated by Jens Funke.

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Miyazaki, T. On Fourier–Jacobi expansions of real analytic Eisenstein series of degree 2. Abh. Math. Semin. Univ. Hambg. 84, 85–122 (2014). https://doi.org/10.1007/s12188-014-0092-8

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  • DOI: https://doi.org/10.1007/s12188-014-0092-8

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