The Investigation On The Properties Of Some Kinds Of Combinatorial Sequences

In this dissertation, the properties on some kinds of classical combinatorial sequences and some identities involving generalized Bernoulli and Euler polynomials are investigated by means of generating function and calculous technique. The main contents of this thesis can be summarized as follows:1. In Chapter 1, we introduce the development of the classical combinatorial sequences and the preparative knowledge.2. In Chapter 2, we utilize two integral formulas of [1] and [7] to discuss the followingthree kinds of infinite sums involving reciprocal of binomial coefficients: (1).Some calculation formulas and recurrence relationsof above sums are derived.3. Bernoulli and Euler polynomials play an important role in number theory and special function. In chapter 3, we study their propertities: (1). the values of binary Bernoulli and Euler poynomials at rational points;(2). some elegant identities of paper [22] are extended to other forms of generalized Bernoulli and Euler polynomials.4. The total number of paths with diagnal steps of joining origin O(0,0) and point (i,j) are called Delannoy number, denoted by d_i,j which satisfies the recurrence: d_i,j = d_(i-1),j + d₍(i,(j-1)) + d_(i-1),(j-1). In Chapter 4, we discuss the special case of d_i,j with i = j: (1). we obtain an integral expression for Delannoy number;(2). we derive a close expression of convolution sums of Delannoy number by means of generalized Fibonacci numbers .5. In Chapter 5, we use a relation between Salie numbers and Euler munbers to discuss the computation of the sums of Salie'numbers as the formuler...