A note on Kronecker's double sum and non-holomorphic Eisenstein series
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摘要: 由狄利克雷特征对克罗内克二重级数的扭曲生成了一族权为k级别为N的非全纯艾森斯坦级数.由此,我们推导出了它的惠特克函数表示以及艾森斯坦级数的泛函方程.Abstract: A family of non-holomorphic Eisenstein series of weight k and level N is generated from twisting of the Kronecker double series by Dirichlet characters and from which we will derive its representation in terms of the Whittaker function and the functional equation for the Eisenstein series.
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Key words:
- Dirichlet character /
- conductor /
- Gaussian sum /
- twisted Poisson summation formula /
- Whittaker function
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[1] BUMP D. Automorphic Forms and Representations[M]. Cambridge:Cambridge University Press, 1997. [2] GOLDFELD D, HUNDLEY J. Automorphic Representations and L-Functions for the General Linear Group (Ⅰ)[M]. Cambridge:Cambridge University Press, 2011. [3] ERDÉLYI A. Higher Transcendental Functions (Ⅰ)[M]. New York: McGraw-Hill, 1953. [4] LANG S. Elliptic functions[M]. 2nd ed. New York: Springer-Verlag, 1987. [5] WEIL A. Elliptic Functions According to Eisenstein and Kronecker[M]. Berlin:Springer-Verlag, 1976. [6] WHITTAKER E T, WATSON G N. A Course of Modern Analysis[M]. 4th ed. Cambridge:Cambridge University Press, 1958.
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