Research On Some Recursive Polynomials

Second-order linear recursive sequences and polynomials play an important role in the study of number theory.The research on various properties has been valued and favored by scholars.Among them,the Fibonacci sequence,the Lucas sequence and the Pell sequence play an important role in the study of some famous number theory problems.For example,in combinatorial mathematics,the generation function of Fibonacci sequence can be used to construct various identities.Many experts and scholars have studied the nature of recursive sequences and polynomials from various directions.On the one hand,scholars have studied the intrinsic properties of polynomials and series.After Japanese scholars first studied the problem of the reciprocal summation of Fibonacci series,more scholars have studied similar problems.The research object extends from the Fibonacci sequence to more sequences,then from the series to the polynomial.Scholars have also discussed the derivative and integral summation of series and polynomial,which has led to the rapid expansion of research in this direction.On the other hand,some scholars have discussed the relationship between different series and different polynomials,such as using the Chbyshev polynomial to study Fibonacci series,Lucas series or other series.Under the guidance of many scholars' different ideas,this paper also discusses a series of polynomials and has obtained some conclusions:The first aspect of this paper is to summarize the research status and history of Lucas polynomial.Based on the previous studies,the integral summation problem of Lucas polynomial is calculated and the related derivation process is carried out.The second aspect of this paper is to study the relationship between two kinds of Chebyshev polynomials,and give some identities about the product sum and product difference of two kinds of Chebyshev polynomials,and the corresponding derivation is proved.The third aspect of this paper mainly constructs a new second-order linear recursive polynomial,and uses the elementary method and combination technique to obtain two theorems,and carries out rigorous mathematical derivation.