关于一类poly-Dedekind DC和的研究  

A study on a type of poly-Dedekind type DC sums

作　　者：马元魁 罗玲玲 KIM Taekyun 李红泽 MA Yuankui;LUO Lingling;KIM Taekyun;LI Hongze(School of Science,Xi’an Technological University,Xi’an 710021,China;Department of Mathematics,Kwangwoon University,Seoul 139-701,Korea;School of Mathematical Sciences,Shanghai Jiao Tong University,Shanghai 200240,China)

机构地区：[1]西安工业大学基础学院,陕西西安710021 [2]光云大学数学系,韩国首尔139-701 [3]上海交通大学数学科学学院,上海200240

出　　处：《西北大学学报（自然科学版）》2023年第3期438-442,共5页Journal of Northwest University（Natural Science Edition）

基　　金：国家自然科学基金(11871317,11926325,11926321)。

摘　　要：研究了Dedekind DC和的一个新的推广——poly-Dedekind DC和。Dedekind DC和中的Euler函数替换成了由polylogarithm函数定义的poly-Euler函数。利用polylogarithm函数的定义、第二类Stirling数的定义、Euler多项式的定义,得到了此类poly-Euler函数满足的一些恒等式,包括此类poly-Euler数与第二类Stirling数之间的关系,以及此类poly-Euler多项式与Euler多项式、第二类Stirling数之间的关系,并证明了此poly-Dedekind DC和满足互反关系。正如经典的Dedekind和一样,可进一步探讨其与模形式、ζ函数以及三角和的关系。As a new generalization of the Dedekind type DC sums,the poly-Dedekind type DC sums are introduced,which are obtained from the Dedekind type DC sums by replacing Euler functions by poly-Euler functions of arbitrary indices arising from polylogarithm functions.By using the definitions of polylogarithm functions,Stirling numbers of second kind,and Euler polynomials,several properties of poly-Euler functions are obtained,including the relations between this type of poly-Euler numbers and Stirling numbers of second kind,the relations between this type of poly-Euler polynomials and Euler polynomials and Stirling numbers of second kind.The poly-Dedekind type DC sums are shown to satisfy a reciprocity relation.It can be further explored in connection with modular forms,ζfunctions,and trigonometric sums,just as in the cases of Apostol-Dedekind sums.

关 键 词：poly-Dedekind DC和 polylogarithm函数 poly-Euler多项式 

分 类 号：O156[理学—数学]