关于Kuznetsov求和公式的注记（英文）  

Note on the Kuznetsov sum formula

作　　者：KANEMITSU S AGARWAL P 李海龙[3] 

机构地区：[1]日本近畿大学信息科学学院 [2]阿南德工程国际学院数学学院,印度拉贾斯坦邦斋浦尔303012 [3]渭南师范学院数学学院,陕西渭南714000

出　　处：《纺织高校基础科学学报》2017年第3期293-301,共9页Basic Sciences Journal of Textile Universities

摘　　要：在自守L-函数理论中,Kuznetsov迹公式是焦点问题之一,它有一些不同的表述方法和形式,其中,最经典的是Iwaniec给出的第一类Fuchs群上的情形.而Motohashi考查了全模群上的情形且与Iwaniec的结果并不相同.在模群关系原理基础上,Ma发现Motohashi的表述是定理2.4的另一种形式,且认为其未考虑Neumann级数.本文就Motohashi利用Selberg将两个Poincaré级数的内积的不同表达形式进行等同处理并推广了定理2.4的方法,这和Iwaniec的结果形式相反,但却符合许多文献中描述的Kuznetsov迹公式.为了使Kuznetsov迹公式的形式更加简单并容易理解,本文利用一些常用的特殊函数详细阐述了马晶与Agarwal的证明过程并证明了一些包含Bessel函数的递推式.In the theory of automorphic L-functions,the Kuznetsov trace formula is one of the highlights.There are a few different statements of the formula.The most comprehensive one for a Fuchsian group of the first kind is given in Iwaniec′s book.In Motohashi′s book,the case of the full modular group is treated and their results look different.In the companion paper by Ma and Agarwal,it is shown on the basis of modular relation principle that the statement of Motohashi is another version of Theorem 2.4 and that it has different outlook lacking the Neumann series.We slightly generalized the method of Selberg adopted by Motohashi of equating the two different expressions for the inner product of two Poincaréseries to deduce Theorem 2.4 below,which is the reversed form according to Iwaniec but it is the form stated as the Kuznetsov trace formula in most of the literature.In this note we shall elaborate the proof in the Ma and Agarwal′s paper using more familiar special functions and give proofs of intermediate formulas involving Bessel functions.This makes the situation more transparent surrouding the Kuznetsov trace formula and make it more accessible.

关 键 词：Kuznetsov和 自守L-函数 Neumann级数 Poincaré级数 BESSEL函数 

分 类 号：O156[理学—数学]