Applications Of Some Operators In Combinatorial Mathematics

It is one of important methods in combinatorics to solve combinatorial problem by operators. In this paper, differential operator and the invertible shift-invariant operator are used to prove some identities about the Bernoulli polynomial and so on. We also introduce a new univariate interpolation operator: Bernoulli-type Grünwald operator. Some properties and the rate of convergence of the new combined operator are studied. The main results of the thesis can be summarized as follows:1. Some operator equations concerning difference operator are given in terms of generating function, and their applications are discussed in combinatorial identities. We also obtain some recurrence relations and identities about the special combinatorial numbers, such as Bernoulli polynomial, Bernoulli number, Stirling number and so on .2. Some identities about the Bernoulli polynomial are introduced by properties of the invertible shift-invariant operator relative to the Bernoulli polynomial.3. A kind of Bernoulli-type Grünwald operator is proposed by a linear combination of Grünwald operator with the generalized Taylor polynomial, the error estimates are also proved.