On Chebyshev Polynomials, Fibonacci Polynomials, and Their Derivatives

Yang Li

doi:10.1155/2014/451953

Abstract

We study the relationship of the Chebyshev polynomials, Fibonacci polynomials, and theirrth derivatives. We get the formulas for therth derivatives of Chebyshev polynomials being represented by Chebyshev polynomials and Fibonacci polynomials. At last, we get several identities about the Fibonacci numbers and Lucas numbers.

Highlights

As we know, the Chebyshev polynomials and Fibonacci polynomials are usually defined as follows: the first kind of Chebyshev polynomials is Tn+2(x) = 2xTn+1(x) − Tn(x) and n ≥ 0, with the initial values T0(x) = 1 and T1(x) = x; the second kind of Chebyshev polynomials is Un+2(x) = 2xUn+1(x)−Un(x) and n ≥ 0, with the initial values U0(x) = 1 and U1(x) = 2x; the Fibonacci polynomials are Fn+2(x) = xFn+1(x) + Fn(x) and n ≥ 0 with the initial values F0(x) = 0 and F1(x) = 1
Falcon and Plaza [4,5,6] presented many formulas and relations between Fibonacci polynomials and their derivatives. This fact allows them to present a family of integer sequences in a new and direct way
We obtain the following theorems and corollaries. These results strengthen the connections of two kinds of polynomials

Summary

Introduction

Zhang [3] used the rth derivatives of Chebyshev polynomials to solve some calculating problems of the general summations. Falcon and Plaza [4,5,6] presented many formulas and relations between Fibonacci polynomials and their derivatives. This fact allows them to present a family of integer sequences in a new and direct way. We obtain the following theorems and corollaries These results strengthen the connections of two kinds of polynomials. They are helpful in dealing with some calculating problems of the general summations or studying some integer sequences. For any positive integers n, m, and r, one has the following identities: imL 2 m (−1)sm+n−k22r−1 (n im (n − k)!

Some Lemmas

Proof of the Theorems and Corollaries