Abstract

We use the combinatorial method and algebraic manipulations to obtain several interesting identities involving the power sums of the derivative of the first kind Chebyshev polynomials. This solved an open problem proposed by Li (2015).

1. Introduction

For any integer , the famous Chebyshev polynomials of the first and second kind and are defined as follows: where denotes the greatest integer .

It is clear that and are the second-order linear recurrence polynomials; they satisfy the recurrence formulae: , , and for all , , , and for all .

The general term formulae of and are Both and are orthogonal polynomials. That is,

About the other various properties of the Chebyshev polynomials, some authors had studied them and obtained many interesting conclusions. For example, Ma and Zhang [1], Wang and Han [2], Cesarano [3], and Lee and Wong [4] proved a series of identities involving Chebyshev polynomials. Bhrawy and others (see [5–10]) obtained many important applications of the Chebyshev polynomials.

Very recently, Li [11] proved some identities involving power sums of and . That is, for any positive integers and , one has the identities:(a) (b) (c) (d) 

As for some applications of these results, Xiaoxue Li obtained some divisibility properties involving Chebyshev polynomials. At the same time, she also proposed the following open problem.

Does there exist an exact expression for the derivative or integral of the Chebyshev polynomials of the first kind in terms of the Chebyshev polynomials of the first kind (and vice versa)? That is to say, does there exist an exact expression for the summations in terms of the Chebyshev polynomials of the first kind?

Does there exist an exact expression for in terms of or where denotes the derivative of with respect to ?

In this paper, as a note of  [11], we give some identities involving the derivative of the first kind Chebyshev polynomials. That is, we will prove the following.

Theorem 1. For any integers and , one has the identity where , , , and are computable constants.

Theorem 2. For any positive integers and , one has the identity where , , , and are computable constants.

For some special , from Theorem 1 with and Theorem 2 with , we can also deduce the following two corollaries.

Corollary 3. For any positive integer , one has the identity

Corollary 4. For any positive integer , one has the identity

2. Two Simple Lemmas

In this section, we will give two simple lemmas, which are necessary in the proofs of our theorems. First we express and in terms of and . That is, we have the following.

Lemma 1. For any positive integers and , one has the identities

Proof. For any positive integer and real number and , by using the familiar binomial expansion we have Now we take in (14); note that ; from the definition of and we have the identities Note that ; from (15) we may immediately deduce This proves Lemma 1.

Lemma 2. Let and be two positive integers. Then for any and , we have the recurrence formula: (I) There exist some computable constants and such that (II) In particular for and , we have the identities: (A) (B) 

Proof. For any positive integer , let ; it is clear that Note that the binomial expansion is as follows: and from (21) we may immediately deduce the identity: This proves the recurrence formula (I).
It is easy to prove (II) by recurrence formula (I) and the complete mathematical induction.
From the recurrence formula (I) and noting that the identity we can also deduce This proves formula (A).
Identity (B) follows from (24), (A), and the recurrence formula (I) with . This proves Lemma 2.

Some Note. The first part of Lemma 2 obtained an interesting recurrence formula for the computation of the summation . But if positive integer is large enough, then the computation of the recurrence formula is more complex, and so we have not given the exact constants and in formula (II).

3. Proofs of the Theorems

In this section, we will complete the proofs of our theorems. First we prove Theorem 1. Let , ; then note that , ; from the definition of and the second formula of Lemma 1 we have Applying formula (II) of Lemma 2 we have Note that ; from the definitions of and we have Combining (27)–(29) we may deduce that From (26) and (30) we have the identity This proves Theorem 1.