Study On The Sums Of Balancing Polynomials And Chebyshev Polynomials

Balancing numbers have been widely studied in recent years.In this paper.the balancing polynomials we study are constructed on the basis of balancing numbers.We find that balancing polynomials are closely related to Chebyshev polynomials.Polynomial summation of various forms has always been the focus of number theory,such as the sums of powers of polynomials,the reciprocal sums of the polynomials,the reciprocal sums of the square of the polynomials,and the reciprocal sums of products of two consecutive polynomials.This paper mainly studies the summation of balancing polynomials and Chebyshev polynomials.and gives the following results1.In this paper,a second order non-linear recursive sequence M(n,i)is studied.By using this sequence,the properties of the power series and the combinatorial methods,some interesting identities of the structural properties of balancing numbers and balancing polynomials are deduced2.In this paper,the reciprocal sums of the balancing polynomials,the reciprocal sums of the square of the balancing polynomials,and the reciprocal sums of products of two consecutive balancing polynomials are studied.Then we apply the floor function to these sums and get several concise equalities involving the balancing polynomials3.In this paper,the combinatorial method and algebraic manipulations are used to study the sums of powers of Chebyshev polynomials and some interesting identities of the structural properties of Chebyshev polynomials are deduced.