关于一类poly-Dedekind DC和的研究

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

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.