【作者】 王鑫；

【导师】 乌云高娃；

【作者基本信息】 内蒙古大学 ， 数学， 2022， 硕士

【摘要】 本文运用概率方法来研究组合序列,利用不同分布的随机变量的高阶矩、独立性以及特征函数,推出广义Stirling数、广义高阶Bernoulli序列的9)阶矩.然后对这些组合序列的矩表示进行变换得到一些新的性质,同时利用矩表示推导了广义Stirling数、广义高阶Bernoulli序列与其他经典组合序列之间的新的组合恒等式.主要工作如下:1.首先介绍了组合数学的历史背景,国内外对组合数学的研究现状,以及组合数学在概率方向的发展情况.然后介绍了广义Stirling数,广义高阶Bernoulli多项式的研究情况,概率论的相关基础知识,以及文章中用到的组合序列的矩表示.2.主要利用服从均匀分布、伽马分布随机变量的9)阶矩和取系数的方法给出广义Stirling数,第一类r-Stirling数的矩表示.然后,对第一类广义Stirling数,第一类r-Stirling数的-analogues的矩表示进行变换,得到了这两类组合序列与第一类Stirling数、高阶Daehee数之间的恒等式.3.主要利用服从Laplace分布的随机变量的9)阶矩、特征函数,给出含参数(6,(7,(8的广义高阶Bernoulli序列的矩表示.然后对高阶Bernoulli序列的矩表示进行变换,验证了高阶Bernoulli序列相关的性质,同时也得到一些新的性质.最后运用概率方法证明了广义高阶Bernoulli数及其多项式的一组对称恒等式,利用这组恒等式得到广义高阶Bernoulli序列与调和数、第二类Stirling数、错排数、Fibonacci数等之间的新的恒等式.更多还原

【Abstract】 In this thesis,the probabilitic method is used to study the combinatorial sequences,and higher order moments,independence and characteristic functions of random variables with different distributions are used.The moment of generalized Stirling number and gen-eralized higher-order Bernoulli sequences are derived.Then the moment representations of these combinatorial sequences are transformed to obtain some new properties,and the moment representations are used to deduce new groups between generalized Stirling numbers,generalized higher-order Bernoulli sequences and other classical combinatorial sequences.The main work is as follows:1.Firstly,we introduces the historical background of combinatorial mathematics,the research status of combinatorial mathematics at home and abroad,and the development of combinatorial mathematics in the direction of probability.Then it introduces the research of generalized Stirling numbers,generalized higher-order Bernoulli polynomials,the relevant basic knowledges of probabilitic theory,and the moment representation of combinatorial sequences used in this paper.2.The moment representation of generalized Stirling numbers and the first kind of r-stirling numbers is given by using the n moment of random variables subject to uniform distribution and gamma distribution and the method of taking coefficients.Then the moment representations of the first kind of generalized Stirling numbers and the-analogues of r-Stirling numbers of the first kind are transformed,and the identities between the two kinds of combinatorial sequences and the first kind of Stirling numbers and higher-order Daehee numbers are obtained.3.Using the n moment and characteristic function of random variables subject to Laplace distribution,the moment representation of generalized higher-order Bernoulli sequences with parameters a,b,c is given.Then the moment representation of higher-order Bernoulli sequences is transformed,the related properties of higher-order Bernoulli sequences are verified,and some new properties are obtained.Finally,a set of symmetric identities of generalized higher-order Bernoulli numbers and their polynomials are proved by using the probabilitic method.Using this set of identities,new identities among generalized higher-order Bernoulli sequences and harmonic numbers,the second kind of Stirling numbers,derangement numbers and Fibonacci numbers,etc are obtained.更多还原