Some Results On Combinatorial Sequences And Related Matrices

The main object of this dissertation is to study some combinatorial sequences and related matrices. The contents can be summarized as follows:Chapter 1 introduces the developments of the theories of combinatorial sequences and matrices, and the following two chapters are the results of the present thesis.Chapter 2 contains discussions on some combinatorial sequences and can be divided into two parts.The first part of Chapter 2 is devoted to the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials. In this part, we obtain two relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials, and give the explicit expressions for the sums of products of the Apostol-Bernoulli polynomials and of the Apostol-Euler polynomials.In the second part of Chapter 2, we have a study on partial exponential Bell polynomials and Sheffer sequences. By making use of the Bell polynomials, two characterizations of Sheffer sequences are presented first, and then by substituting associated sequences and Sheffer sequences for the variables x₁,X₂,…of the Bell polynomials, many general identities are derived.Chapter 3 aims to study the matrices related to some combinatorial sequences, and there are two parts in this chapter also.The first part of Chapter 3 gives the factorizations of the Bell matrices and the iteration matrices, which provide unified approaches to the factorizations of many lower triangular matrices in combinatorics.In the second part of Chapter 3, we establish the theory of generalized Riordan arrays. Besides the basic properties, the relationships between generalized Riordan arrays and generalized Sheffer sequences are studied, and it is shown that the Riordan group and the group of Sheffer sequences are isomorphic. Based on this fact, from the Sheffer sequences, many special Riordan arrays are obtained. Moreover, the recurrence relations satisfied by the elements of the Riordan arrays, the factorizations of the Riordan arrays, the inverse relations problem and the connection constants problem are also discussed.