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# Some new identities of Chebyshev polynomials and their applications

*Advances in Difference Equations*
**volume 2015**, Article number: 355 (2015)

## Abstract

In this paper, we use the properties of Chebyshev polynomials, elementary methods, and combinational techniques to study the computational problem of one kind of convolution sums involving second kind Chebyshev polynomials, and we give an exact computational method, which expresses the sums as second kind Chebyshev polynomials. As some applications of our results, we also obtain several new identities and congruences involving the second kind Chebyshev polynomials, Fibonacci numbers, and Lucas numbers.

## 1 Introduction

For any integer \(n\geq0\), the famous Chebyshev polynomials of the first and second kind \(T_{n}(x)\) and \(U_{n}(x)\) are defined as follows:

and

where \([m]\) denotes the greatest integer ≤*m*.

It is clear that \(T_{n}(x)\) and \(U_{n}(x)\) are the second-order linear recurrence polynomials, they satisfy the recurrence formulas

The general formulas of \(T_{n}(x)\) and \(U_{n}(x)\) are

and

The generating functions of \(T_{n}(x)\) and \(U_{n}(x)\) are

and

As regards the elementary properties of Chebyshev polynomials, some authors had studied them, and they obtained many interesting conclusions. For example, Zhang [1] proved that for any positive integer *k* and nonnegative integer *n*, one has the identity

where \(U^{(k)}_{n}(x)\) denotes the *k*th derivative of \(U_{n}(x)\) with respect to *x*, the summation is taken over all \(k+1\)-dimension nonnegative integer coordinates \((a_{1}, a_{2}, \ldots, a_{k+1})\) such that \(a_{1}+a_{2}+\cdots+a_{k+1}=n\).

As some applications of (3), Zhang [1] obtained some identities involving Fibonacci numbers and Lucas numbers.

Ma and Zhang [2], Li [3], Wang and Zhang [4], Cesarano [5], Lee and Wong [6] also proved a series of identities involving Chebyshev polynomials. Bhrawy *et al.* (see [7–10]) and Bircan and Pommerenke [11] obtained many important applications of the Chebyshev polynomials. For an overview of some new work related to the generating functions of Chebyshev polynomials of the first and the second kind, one may refer to Cesarano [12].

It is clear that an interesting problem is whether one can express \(U^{(k)}_{n+k}(x)\) by the second kind Chebyshev polynomials.

It seems that none had studied this problem yet, at least we have not seen any related result before. The problem is interesting and important, because it can reveal the inner relations of the second kind Chebyshev polynomials, and it can also express a complex sum in a simple form.

This paper, as a note of [1], we give an exact computational method, which express \(U^{(k)}_{n+k}(x)\) by the second Chebyshev polynomials. That is, we shall prove the following main conclusion.

### Theorem

*For any positive integer*
*k*
*and nonnegative integer*
*n*, *we have the identity*

*where*
\((1-x^{2})U'_{n}(x)=(n+1)U_{n-1}(x)-nxU_{n}(x)\).

It is clear that this theorem gives an exact computational method, which expresses \(U^{(k)}_{n+k}(x)\) by Chebyshev polynomials \(U_{n}(x)\). From this theorem we may immediately deduce the following.

### Corollary 1

*For any positive integers*
\(n\geq k\geq2\), *we have the identity*

*where*
\(R(n,k,x)\)
*and*
\(S(n,k,x)\)
*are two computable polynomials of*
*n*, *k*, *and*
*x*
*with integral coefficients*.

Especially for \(k=2\) and 3, we have the following.

### Corollary 2

*For any nonnegative integer*
*n*, *we have the identity*

### Corollary 3

*For any nonnegative integer*
*n*, *we have the identity*

It is clear that the left-hand side of (3) is a polynomial of *x* with integral coefficients, so from Corollary 2 and Corollary 3 we can also deduce the following.

### Corollary 4

*For any nonnegative integer*
*n*, *we have the congruence*

### Corollary 5

*For any nonnegative integer*
*n*, *we have the congruence*

As some applications of our results, we find that there are some close relationships among the Chebyshev polynomials, Fibonacci numbers \(F_{n}\), and Lucas numbers \(L_{n}\). These sequences are defined as

and

for all integers \(n\geq0\).

It is clear that they also satisfy the second-order linear recurrence formulas \(F_{n+2}=F_{n+1}+F_{n}\), \(L_{n+2}=L_{n+1}+L_{n}\) for all \(n\geq0\) with \(F_{0}=0\), \(F_{1}=1\), \(L_{0}=2\), \(L_{1}=1\). Some papers related to Fibonacci numbers and Lucas numbers can also be found in [13–17]. From our results we can also deduce the following identities.

### Corollary 6

*For any positive integers*
*m*
*and*
*n*, *we have the identity*

### Corollary 7

*For any positive integers*
*m*
*and*
*n*, *we have the identity*

Taking \(m=1\) in Corollaries 4 and 5 we may immediately deduce the following.

### Corollary 8

*For any nonnegative integer*
*n*, *we have the identities*

*and*

## 2 Several simple lemmas

In this section, we shall give several simple lemmas, which are necessary in the proofs of our results. First of all we have the following.

### Lemma 1

*For any positive integers*
\(n\ge k> 0\), *we have the identity*

### Proof

It is clear that the second kind Chebyshev polynomials \(U_{n}(x)\) satisfy the differential equation

So for any positive integer \(n\geq k>2\), we have

Differentiating (4) repeatedly \((k-2)\) times we obtain

or

This proves Lemma 1. □

### Lemma 2

*For any positive integers*
\(n\geq k\geq1\), *we have the identity*

*where*
\(R(n,k,x)\)
*and*
\(S(n,k,x)\)
*are two computable polynomials of*
*n*, *k*, *and*
*x*
*with integral coefficients*.

### Proof

We prove Lemma 2 by complete induction. Note that we have the identity

or

So Lemma 2 holds for \(k=1\).

Assume that Lemma 2 holds for all positive integers \(1\leq k\leq m\). That is, for all positive integers \(1\leq k\leq m\), we have

Then for \(k=m+1\), from (5), (6), and Lemma 1 we have

This proves Lemma 2 by complete induction. □

### Lemma 3

*For any positive integers*
*m*
*and*
*n*, *we have the identities*

### Proof

See Lemma 3 in Zhang [1]. □

## 3 Proof of the theorem

In this section, we shall complete the proofs of our all results. It is clear that our theorem follows from (3) and Lemma 1. In fact, substituting *n* by \(n+k\) in Lemma 1 we have

Combining identities (3) and (7) we may immediately deduce

This proves our theorem.

It is clear that Corollary 1 follows from our theorem and Lemma 2.

Now we prove Corollary 2. Taking \(k=2\) in our theorem and noting that \((1-x^{2})U'_{n}(x)=(n+1)U_{n-1}(x)-nxU_{n}(x)\) we have

This proves Corollary 2.

To prove Corollary 3, taking \(k=3\) in our theorem we have

This proves Corollary 3.

Now we prove Corollary 6. Taking \(x=\frac{3}{2}\) in (1) and (2), we note the identities

and

Applying Lemma 3 and (9) we also have

Taking \(x=T_{m} (\frac{3}{2} )\) in Corollary 2, applying (8), (9), and (10) we have

or

where we have used the identities \(F_{m}\cdot L_{m}=F_{2m}\) and \(L^{2}_{2m}-4=5\cdot F^{2}_{m} \).

Similarly, taking \(x=T_{m} (\frac{3}{2} )\) in Corollary 3, from (8), (9), and (10) we can also deduce the identity

This proves Corollaries 6 and 7.

Corollary 8 follows from Corollary 7 with \(m=1\), \(L_{2}=3\), \(F_{1}=F_{2}=1\), \(F_{4}=3\).

This completes the proofs of our all results.

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## Acknowledgements

The author would like to express gratitude to the editors and anonymous reviewers for their valuable suggestions, which improved the presentation of this paper. This work is supported by the N.S.F. (11371291) of P.R. China.

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The author declares that they have no competing interests.

### Author’s contributions

WS obtained the main result and completed all the parts of this manuscript. WS read and approved the final manuscript.

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Siyi, W. Some new identities of Chebyshev polynomials and their applications.
*Adv Differ Equ* **2015**, 355 (2015). https://doi.org/10.1186/s13662-015-0690-5

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DOI: https://doi.org/10.1186/s13662-015-0690-5