Summation Of Generalized Fibonacci Numbers And Its Application

The property and application of Fibonacci numbers are a hot topic in elementary number theory.Its research results have important theoretical significance and application value in mathematics and other disciplines(such as physics,economics,biology).In recent years,many experts and scholars have studied and promoted the Fibonacci numbers from different angles.In this paper,we mainly study the summation of generalized Fibonacci numbers and its application.The paper is divided into four chapters,and the specific arrangements are as follows:In chapter 1,we extend the Fibonacci numbers by changing the recurrence relation and retaining the form of initial value.By using the elementary method and the properties of the integral function,we study the summation of the generalized Fibonacci numbers.We give the calculation formulas of the finite sum of the reciprocal of the continuous subsequence of the generalized Fibonacci numbers,the finite sum of the reciprocal of the square of the generalized Fibonacci numbers,and the finite sum of the reciprocal of the product of two terms of the generalized Fibonacci numbers.These identities reveal the relationship between the rounding part of the reciprocal finite sum of the generalized Fibonacci numbers and some terms of the sequence itself.In chapter 2,on the basis of the previous chapter,we do further research.We extend the finite sum of the reciprocal of generalized Fibonacci numbers to the alternating finite sum.Then we give the formulas for computing the interleaved finite sum of the reciprocal of the continuous subsequence of the generalized Fibonacci numbers and the interleaved finite sum of the reciprocal of the product of two terms.In chapter 3,the Fibonacci numbers is mainly promoted by changing the form of the initial value and the recurrence relationship at the same time.Then the application of the generalized Fibonacci numbers in the Diophantine equation is studied,that is,the generalized Fibonacci numbers is used to characterize the positive integer solutions of some special Diophantine equations.Firstly,the famous Lucas-balancing and Lucas-cobalancing numbers are introduced and their properties are studied.These two numbers are special cases of generalized Fibonacci numbers.Secondly,by using the pell equation method and recursive sequence method in elementary number theory,it is proved that some special Diophantine equations are all positive integer solutions with the Lucas-balancing and Lucas-cobalancing numbers.In chapter 4,we summarize the achievements and look forward to the work.We summarize the main contents of this paper and put forward some problems for further study.