Application Of The Probability Method In Proving Combinatorial Identities

In this paper, we apply the methods and skills of probability theory and generating function in combinatorial analysis systematically, and investigate generalized harmonic numbers and generalized Bernoulli polynomial. We obtain their moments express and some new identities about derangement numbers, bell numbers, Bernoulli numbers, Euler numbers, Harmonic numbers, the second kind Stirling numbers. Showing that probability method is of great importance in proving combinatorial identities.The main works are as follows:Chapter 2:First, we introduce the definition of the generalized Harmonic numbers Hn(Ï„). The moments of common random variables (derangement numbers, bell numbers. Bernoulli numbers, Euler numbers, harmonic numbers, the second kind Stirling numbers) are represented in combinatorial identities. By means of the probability method we obtain some identities of generalized harmonic numbers Hn(Ï„)Chapter 3:We introduce the definition of the generalized Bernoulli polynomials Bn(x;a,b,c). By means of the probability method and generating function, we obtain properties of the generalized Bernoulli numbers and the moment express of the gener-alized Bernoulli polynomials Bn(x:a,b,c) and some identities of generalized Bernoulli polynomial B(x:a,b,c).