The Study On Properties And Applications Of Second-order Linear Recursive Polynomials

Fibonacci polynomials,Lucas polynomials,Fibonacci numbers,Lucas numbers,etc.are the most common second-order linear recursive polynomials and numbers in the field of number theory.Their arithmetical properties play a role in economics,physics,and science.The important role,so the study of second-order recursive polynomials and their corresponding sequence has been the basis and focus of the work of number theory.Japanese scholars Ohtsuka and Nakamura used the relationship of inequalities ingeniously and found the reciprocal of Fibonacci sequence and the formula for rounding,but this method is not generalizable.Subsequently,many scholars launched more general research,such as other second-order linear recursion sequences,polynomial reciprocal summation formulas,and higher-order infinite sum reciprocal formula calculations.Until now,scholars have devoted themselves to the study of many arithmetic properties of second-order linear recursive polynomials such as Fibonacci polynomials,two types of Chebyshev polynomials,and Lucas polynomials,including lower power formulas of higher powers,convolutional Simple calculation formula,calculation of integral sum,etc.,and get many interesting conclusions.In this paper,by using the expression and properties of the polynomial generating function,the convolution and combination formulas of various polynomials are calculated,and the related research is gradually deepened.Using the elementary calculation method,the convolution and higher order of some second-order linear recursive polynomials are studied.Calculation of power sum and related reciprocal sum.In fact,by taking some special values in the second-order linear recursive polynomials,you can immediately get some special calculation formulas for the second-order linear recursive sequences,which provides a more general method for studying the identities of the second-order linear recursive sequences Method,so it is necessary to study.In addition,the research on the related properties of arithmetic functions in analytic number theory is also a hot topic in number theory research.In this paper,the analytical method is used to study the identity of a new sum related to Dedekind sum with the help of the property of Gaussian sum,and its value at a particular point is given.Chapter 2 Since the convolution,sum of high powers,and reciprocal sum of polynomials can be expressed by the simplest calculation formula,that is,the abstract difficult formula is most simplified,therefore,the convolution,high The identities of sums of powers and sums of reciprocals result in several theorems.The third chapter mainly uses analytical methods to discover the computational properties of the new sums related to Dedekind and it can be transformed into Dedekind sum under special conditions.It explains the internal relationship between the two and solves the problems under special values.The identity of the new type of Japanese.