Applications Of The Generating Function Method In Combinatorial Identities

It is one of important methods in combinatorics to prove identities and solve recurrence relations by using generating function. In this paper, generating function is used to study recurrence relations and identities on Stirling numbers, Bernoulli numbers and Bell numbers. The main results of the thesis can be summarized as follows:1. Firstly generating function's properties of unsigned Stirling numbers of first kind and the central factorial numbers are discussed. Secondly generating function is applied to the research of boundary value problems and three term boundary value problems is generalized toκ-term boundary value problems. Lastly, two examples are given to illustrate the obtained method.2. The Stirling numbers of second kind are applied to the research of the generalized central factorial numbers'congruence relations.3. The Bernoulli numbers and the Stirling numbers of two kinds are used as bridge to link two series and discuss the generating functions of the two series.