On the Chebyshev polynomials and some of their new identities

Dong Han ,Xingxing Lv

doi:10.1186/s13662-020-02550-y

Abstract

The main purpose of this paper is, using the elementary methods and properties of the power series, to study the computational problem of the convolution sums of Chebyshev polynomials and Fibonacci polynomials and to give some new and interesting identities for them.

Highlights

For any integer n ≥ 0, the famous Chebyshev polynomial of the first kind {Tn(x)} is defined as follows: Tn+2(x) = 2xTn+1(x) – Tn(x) for all integers n ≥ 0, with T0(x) = 1, T1(x) = x. √Let α = x + x2 – 1 and β = x – x2 – 1 be two characteristic roots of the equation λ2 –2xλ + 1 = 0, we have, αn+1 – βn+1 Un(x) = δ – β, n = 0, 1, 2, . . . , (1)where {Un(x)} is Chebyshev polynomial of the second kind with U0(x) = 1 and U1(x) = 2x
Theorem 1 established an identity for the convolution sums of Chebyshev polynomials of the first kind
The methods adopted in this paper have some good reference for further study of the properties of general second-order linear recursive sequences

Summary

Introduction

For any integer n ≥ 0, the famous Chebyshev polynomial of the first kind {Tn(x)} is defined as follows: Tn+2(x) = 2xTn+1(x) – Tn(x) for all integers n ≥ 0, with T0(x) = 1, T1(x) = x. √. Let α = x + x2 – 1 and β = x – x2 – 1 be two characteristic roots of the equation λ2 –. Where {Un(x)} is Chebyshev polynomial of the second kind with U0(x) = 1 and U1(x) = 2x. The generating functions of Tn(x) and Un(x) are.

Objectives

Findings

Conclusion