Application Of Asymptotic Enumeration Method And Infinite Matrix Theory In Combinatorics

There are many methods studying combinatorial enumeration, and asymptotic enu-meration method and matrix method is one of the most important approaches, respec-tively. In this thesis, we investigate problems of combinatorial enumeration by applying asymptotic enumeration method and matrix method. The contents of this thesis can be summarized as follows:In Chapter 1. we recall the development of asymptotic enumeration method and Euler-Seidel infinite theory, and the following chapters are the main results of this thesis.In Chapter 2, we investigate properties of Hyperfibonacci sequence and Hyperlucas sequence. We establish some identities related to hyperfibonacci numbers and Hyperlucas numbers by means of the generating function method. Further more, we give the asymp-totic values of certain sums involving Hyperfibonacci numbers and Hyperlucas numbers by using Darboux's Theorem. In addition, we compare the accurate values with the asymtotic values.In Chapter 3, we investigate the computation for asymptotic values of sums. We give the asymptotic values of certain sums by Bender's method.