The Study On Some Polynomials And Recurrence Sequences

Polynomials and some recursive sequences have been deeply loved by many scholars in the study of number theory.Especially the Fibonacci sequence,Lucas sequence,Pell sequence and Fibonacci polynomial,Lucas polynomial and Chebyshev polynomial.In recent years,experts and scholars have studied these polynomials and recurrence formulas from various aspects,and have got many interesting results.On the one hand,study the own property of polynomials and recursive sequence.Since Japanese scholars have studied the the infinite sum of reciprocal of the Fibonacci sequence.Then many scholars begin to study the infinite sum of reciprocal of polynomials and some recursive sequences.And obtained many interesting results.In recent years some scholars studied the reciprocal products of generalized Fibonacci numbers,also abtained many interesting results.On the other hand,many scholars study the relationship between polynomial and recursive sequence or other polynomials,and give our many interesting identities.This paper is inspired by previous results,and gives the following conclusions.The first aspect of this paper is to study the infinite product of reciprocal of the Pell sequence and Lucas polynomials.By using the elementary methods and some properties of the Pell sequence and the Lucas polynomials,we convert the infinite problems to the finite problems,and narrow the scope of our proof.Then through the scaling method,we can get our result.The second aspects of this paper,we use an elementary method to study the relationship between Chebyshev polynomials of the first and second kind and Lucas polynomials.We mainly use the way of integral transformation to get the interesting conclusions.