The Generating Function And The Combinatorial Sequences

We intend to investigate some combinatorial sequences using the generating function and derive some valuable combinatorial identities in this paper. The content is as follows:Chapeter 1 is the introduction of this paper, which reviews the problem and the background of the generating function and the classical combinatorial sequences.In chapeter 2, a generalized Fibonacci sequence is defined,and a determinant formula for the generalized Fibonacci sequence is given. Using the theory of Gegen-bauer polynomials, some identities for the generalized Fibonacci sequence are ob-tained.Using generating function, the relationship between generalized Fibonacci sequence and the second kind of Chebyshev polynomials are established. The similar properties for the generalized Lucas sequence are also derived.In chapeter 3,we construct a sequence of Fibonacci-Hessenberg matrices con-taining two paraments,and generalize the some related results on this subjecct.Chapeter 4 disscusses two kinds of Bessel numbers, we disscuss two kinds of Bessel numbers.Generating functions for two kind of Bessel numbers and Bessel inversion formula are obtained.According to the recursion of the Bessel number of the second kind,we obtain the decomposition of the Bessel matrix.