Study On The Properties And Identities Of Some Recurrence Polynomials

In the study of number theory,polynomials and recursive sequences have always been popular among scholars,especially two kinds of Chebyshev polynomials,Fibonacci polynomials,Legendre polynomials,Lucas polynomials and so on,which play an extremely important role in the study of polynomials.In recent years,many experts and scholars have studied the finite product sum of polynomials as one of the hot issues in number theory,and obtained many interesting identities.But in the process of research,when the number is too large,it will be difficult to calculate.This paper takes this as a starting point,using the analysis method and properties of the first kind of Chebyshev polynomials,studies the finite product sum of Legendre polynomials,and gives some new interesting identities,so as to improve the existing results.At the same time,the new identities are extended to the orthogonality of multiple functions and analytic number theory.The conclusions are as followsIn the second chapter,we study the finite product sum of Legendre polynomials.Based on the generating function of Legendre polynomials,the finite product and new identity of Legendre polynomials are obtained by using the knowledge of power series.These studies improve the research of Legendre polynomial finite product sum,and solve the problem that the number of recursive polynomials is too large to calculate.In the third chapter,we study the upper bound estimation of Legendre polynomials.Through the new identities in Chapter 2 and the orthogonality of the first kind of Chebyshev polynomials,the orthogonality of the finite product sum of Legendre polynomials is revealed,which is also the direct application of the new identities in the analytic number theory and the orthogonality of functions in Chapter 2.These studies are of great significance in analytic number theory,and also make new contributions to the study of Gauss sum.The fourth chapter mainly studies Fibonacci polynomial F_The recurrence relation and general term formula of n(x)are given,and Fibonacci polynomials and Lucas polynomials L are established_Then the identity of the product sum of Fibonacci polynomials is obtained.