华东师范大学学报（自然科学版） >

2018 , Vol. 2018 >Issue 3: 1 - 17

DOI: https://doi.org/10.3969/j.issn.1000-5641.2018.03.001

数学

关于克罗内克二重和与非全纯艾森斯坦级数的一个注记

  • 沈力健
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  • 佛罗里达大学 数学系, 佛罗里达 盖恩斯维尔 32611-8105, 美国
沈力健,男,教授,研究方向为函数论.E-mail:shen@ufl.edu.

收稿日期: 2017-09-11

  网络出版日期: 2018-05-29

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A note on Kronecker's double sum and non-holomorphic Eisenstein series

  • SHEN Li-chien
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  • Department of Mathematics, University of Florida, Gainesville FL 32611-8105, USA

Received date: 2017-09-11

  Online published: 2018-05-29

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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.

关键词： 狄利克雷特征; 导子; 高斯和; 扭曲的泊松求和公式; 惠特克函数

本文引用格式

沈力健 . 关于克罗内克二重和与非全纯艾森斯坦级数的一个注记[J]. 华东师范大学学报（自然科学版）, 2018 , 2018(3) : 1 -17 . DOI: 10.3969/j.issn.1000-5641.2018.03.001

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.

Key words： Dirichlet character; conductor; Gaussian sum; twisted Poisson summation formula; Whittaker function

参考文献

[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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