Application Of Probabilistic Methods In Combinatorial Sequences

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.