关于多重zeta函数及Bernoulli数的若干恒等式

【作者】 王冰；

【导师】 徐哲峰；

【作者基本信息】 西北大学 ， 基础数学， 2013， 硕士

【摘要】 多重zeta函数又称Euler-Zagier和,因其在量子力学、扭结理论、上同调理论等不同学科分支上的应用,而得到了国内外学者的密切关注和重视.通过众多学者的深入研究,得到了一系列重要的研究成果.在对多重zeta函数研究的过程中,用到了Bernoulli数、harmonic shuffle关系、对称函数以及其它一些数学知识.这里我们介绍一下Bernoulli数.Bernoulli数是Jacob Bernoulli在研究形如1κ+2κ+3κ+…+(n-1)κ的和式时引入的.事实上,Bernoulli数是Bernoulli多项式当x=0时的特殊情况.Bernoulli数和Bernoulli多项式有许多重要性质,是人们研究其它问题的有力工具.之后,又有学者提出了广义Bernoulli数、广义Bernoulli多项式的定义.有关这方面的研究和应用,是数论研究的重点之一.基于上述背景,本文主要研究如下几方面的问题：1.利用Bernoulli数以及harmonic shuffle关系研究多重zeta函数的加权均值,主要研究形如∑f(m,n)ζ((2m,2n-2m)的和式,这里f(m,n)为关于整数m,n的函数.2.利用初等方法研究Bernoulli数、Euler数以及广义n阶Bernoulli数、广义n阶Euler数,对某些已有结论进行了推广,除此之外还得出了若干新的关系式.更多还原

【Abstract】 Multiple zeta function, also known as Euler-Zagier sum, has been kept close watch on and attached importance to by domestic and overseas scholars because of its application in different subdisciplines such as quantum mechanics, knot theory, cohomology theory and so on. A series of important research results have been obtained due to the intensive study made by numerous researchers.Knowledge of Bernoulli numbers, harmonic shuffle relation, symmetric func-tion need to be used during the process of researching multiple zeta function. Here we introduce Bernoulli numbers.Bernoulli numbers was introduced by Jacob Bernoulli when he studied the summation form like lκ+2κ+3κ+…+(n-1)κ. In fact, Bernoulli numbers is the special case when x equals0in Bernoulli polynomial. Bernoulli numbers and Bernoulli polynomial, which are powerful means for people to research other problems, have many important properties.Generalized Bernoulli polynomial was presented by several researchers later and the research on and application of it is one of the important aspects that number theory should focus on.Based on the above background, this paper mainly studies the following problems:1. The study of the weighted mean of multiple zeta function, namely the summation form like Σ f(m, n)ζ(2m,2n-2m), has been made by Bernoulli numbers and harmonic shuttle relation.2. Elementary method has been used to study Bernoulli numbers, Euler numbers, generalized n-th Euler numbers and generalized n-th Bernoulli num-bers. Some existing conclusions have been generalized and some new relationship have been obtained in this paper.更多还原