庞加莱级数的双曲傅里叶系数
Hyperbolic Fourier coefficients of Poincaré series
摘要：
庞加莱在1911年和彼得森在1932年给出了现在经典的表达式抛物傅里叶系数的全纯庞加莱级数的贝塞尔函数和Kloosterman总和。后来，在1941年，彼得森介绍了双曲和椭圆傅立叶展开的模形式和相关的双曲和椭圆庞加莱系列。本文用超几何级数和Good的广义Kloosterman和表示抛物型和双曲型Poincaré级数的双曲傅里叶系数.在一个明确的例子为模群，我们看到，双曲Kloosterman和对应于一个总和格点上的双曲线包含在一个椭圆。这允许双曲傅里叶系数的数值计算。
英文摘要：
Poincaré in 1911 and Petersson in 1932 gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincaré series in terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson introduced hyperbolic and elliptic Fourier expansions of modular forms and the associated hyperbolic and elliptic Poincaré series. In this paper we express the hyperbolic Fourier coefficients of Poincaré series, of both parabolic and hyperbolic type, in terms of hypergeometric series and Good's generalized Kloosterman sums. In an explicit example for the modular group, we see that the hyperbolic Kloosterman sum corresponds to a sum over lattice points on a hyperbola contained in an ellipse. This allows for numerical computation of the hyperbolic Fourier coefficients.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
