Some Identities Involving The Fibonacci And Lucas Numbers

The research of combinatorial identity is one of the main subjects in combinatorics. Combinatorial identity is widely applied to the computation in probabilistic statistics, the solution in theoretical physics, and the analysis in algorithm complexity of computer and so on. In combinatorial number theory, the problem of identities involving the Fibonacci and Lucas numbers is thorough and far back. In recent years, the study of the generalized Fibonacci and Lucas numbers is especially developed. There are many methods to study combinatorial identity, of which the generating function is basic and important. The generating function is useful to solve discrete problems and it is a bridge to connect discrete mathematics with continuous analysis. By means of the generating function, it is easy to deduce recursive relation, seek the mean value of the sequence, and prove unimodality and so on. with the generating function, we can also not only prove identities, but discover new identities. Linear homogeneous recurrence sequence with constant coefficient (is also called Fibonacci-Lucas sequence, or F-L sequence) is usually studied as a tool in combinatorial enumeration. However, its fine properties in number theory attract many mathematicians. From 1960s, with people's interest increasing, the sequence has bocome an issue. Of the F-L sequence, the second order one is most mature and has many graceful properties. In this thesis, we consider the second order F-L sequence to establish many identities involving the generalized Fibonacci and Lucas numbers and obtain some new congruences by using the generating function and integration.1. The first chapter of which introduces several symbols, definations and some results on the generalized Fibonacci and Lucas numbers.2. In the second chapter, by using the ordinary generating function, the exponential generating function, the generating function on {(_k~n)} and integration, we present some identities involving the generalized Fibonacci number and get some new congruences.3. In the third chapter, by using the similar methods to the last chapter, we give some identities involving the generalized Lucas number and new congruences.