The Identitiy Of Bell Polynomial With Its Application

In combinatorial mathematics,there are a great many of specific sequences. Such as binomial coefficients, Fibonacci numbers, Lah numbers, Bernoulli numbers, Euler numbers, Stirling numbers and Bernoulli polynomials, Euler polynomials, Bell polynomials and so on. These special sequences can satisfy large numbers of identities, and are widely used in combinatorial mathematics, number theory, numerical analysis and other fields. The research of these special sequences which has great significance, has always been one of the main topics in combinatorial mathematics.In this paper, using the generating function and Riordan matrix methods, we study the relationships between Bell polynomials and other combinatorial sequences, and get some combinatorial identities. The contents can be summarized as follows:In Chapterâ… , we introduce the background of Bell polynomials and one important application of Bell polynomials:Faa di Bruno formula, then gives the concept of the generalized Bell polynomials, at last gives the research situations and some significant conclusions in and abroad our country.In Chapterâ…¡, using the method of Riordan matrice, we study the relationships between Bell polynomials and some kinds of combinatorial sequences, including Fibonacci numbers, Harmonic numbers, Genocchi numbers, Bernoulli numbers, Cauchy number etc., and get some identities between Bell polynomials and these sequences.Chapterâ…¢can be divided into four parts. In the first part, we give the concept of convolution polynomials; In the second part, using the method of generating function, we study the relationships between ordinary Bell polynomials and convolution polynomials, and get several identities; In the third part, using the method of derivation we also get some identityies between ordinary Bell polynomials and convolution polynomial. In the last part, using the method of Riordan matrices, we further study the convolution polynomials.