Some new identities of Chebyshev polynomials and their applications

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.

Siyi Advances in Dierence Equations (2015) 2015:355 DOI 10.1186/s13662-015-0690-5

R E S E A R C H Open Access

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Web End = Some new identities of Chebyshev polynomials and their applications

Wang Siyi*

*Correspondence: mailto:[email protected]

Web End [email protected] School of Mathematics, Northwest University, Xian, Shaanxi, P.R. China

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.

MSC: 11B39

Keywords: second kind Chebyshev polynomials; Fibonacci number; Lucas number; identity

1 Introduction

For any integer n , the famous Chebyshev polynomials of the rst and second kind Tn(x) and Un(x) are dened as follows:

Tn(x) = n

[ n ]

[summationdisplay]

k=

()k (n k )!k!(n k)! (x)nk

and

()k (n k)!k!(n k)!(x)nk,

where [m] denotes the greatest integer m.

It is clear that Tn(x) and Un(x) are the second-order linear recurrence polynomials, they satisfy the recurrence formulas

T(x) = , T(x) = x and Tn+(x) = xTn(x) Tn(x) for all n ,

U(x) = , U(x) = x and Un+(x) = xUn(x) Un(x) for all n .

The general formulas of Tn(x) and Un(x) are

Tn(x) =

[bracketleftbig][parenleftbig]x +

Un(x) =

[ n ]

[summationdisplay]

k=

x [parenrightbig]n + [parenleftbig]x x [parenrightbig]n[bracketrightbig] ()

2015 Siyi. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Siyi Advances in Dierence Equations (2015) 2015:355 Page 2 of 8

and

Un(x) =

x [parenrightbig]n+ [parenleftbig]x x [parenrightbig]n+[bracketrightbig]. ()

The generating functions of Tn(x) and Un(x) are

xt xt + t =

n=

x [bracketleftbig][parenleftbig]x +

[summationdisplay] Tn(x)tn [parenleftbig]|x| < , |t| < [parenrightbig]

and

xt + t =

[summationdisplay] Un(x)tn [parenleftbig]|x| < , |t| < [parenrightbig].

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

[summationdisplay]

a+a++ak+=n

Ua(x) Ua(x) Uak+(x) =

n=

U(k)n+k(x), ()

where U(k)n(x) denotes the kth derivative of Un(x) with respect to x, the summation is taken over all k+-dimension nonnegative integer coordinates (a, a, . . . , ak+) such that a +a +

+ ak+ = n.

As some applications of (), Zhang [] obtained some identities involving Fibonacci

numbers and Lucas numbers.

Ma and Zhang [], Li [], Wang and Zhang [], Cesarano [], Lee and Wong [] also proved a series of identities involving Chebyshev polynomials. Bhrawy et al. (see []) and Bircan and Pommerenke [] 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 rst and the second kind, one may refer to Cesarano [].

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 [], 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

[summationdisplay]

a+a++ak+=n

Ua(x) Ua(x) Uak+(x)

=

k k!

U(k)n+k(x) =

k k!

(k )x

x U(k)n+k(x)

k k!

n + n(k + ) + k

x U(k)n+k(x)

[bracketrightbigg],

where ( x)U n(x) = (n + )Un(x) nxUn(x).

Siyi Advances in Dierence Equations (2015) 2015:355 Page 3 of 8

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

Corollary For any positive integers n k , we have the identity

[summationdisplay]

a+a++ak+=n

=

k k! ( x)k

Ua(x) Ub(x) Uc(x)

(n + )(n + ) (n + )(n + )x

( x) Un+(x).

Corollary For any nonnegative integer n, we have the identity

[summationdisplay]

a+b+c+d=n

Ua(x) Ua(x) Uak+(x)

R(n, k, x) Un+k(x) + S(n, k, x) Un+k(x)[bracketrightbig],

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

Especially for k = and , we have the following.

Corollary For any nonnegative integer n, we have the identity

[summationdisplay]

a+b+c=n

= (n + )x

( x) Un+(x)

Ua(x) Ub(x) Uc(x) Ud(x)

= (n + )((n + n + )x (n + n + ))

( x) Un+(x)

x(n + )((n + n + )x (n + n ))

( x) Un+(x).

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

Corollary For any nonnegative integer n, we have the congruence

(n + )xUn+(x) (n + )[parenleftbig]n + (n + )x[parenrightbig]Un+(x) [parenleftbig]mod[parenleftbig] x[parenrightbig][parenrightbig].

Corollary For any nonnegative integer n, we have the congruence

(n + )[parenleftbig][parenleftbig]n + n + [parenrightbig]x [parenleftbig]n + n + [parenrightbig][parenrightbig]Un+(x)

x(n + )[parenleftbig][parenleftbig]n + n + [parenrightbig]x [parenleftbig]n + n [parenrightbig][parenrightbig]Un+(x) [parenleftbig]mod[parenleftbig] x[parenrightbig][parenrightbig].

As some applications of our results, we nd that there are some close relationships among the Chebyshev polynomials, Fibonacci numbers Fn, and Lucas numbers Ln. These

Siyi Advances in Dierence Equations (2015) 2015:355 Page 4 of 8

sequences are dened as

Fn =

[bracketleftbigg][parenleftbigg] +

n

n

[parenleftbigg]

[bracketrightbigg]

and

Ln = [parenleftbigg] +

n

+ [parenleftbigg]

n

,

for all integers n .

It is clear that they also satisfy the second-order linear recurrence formulas Fn+ = Fn+ +

Fn, Ln+ = Ln+ + Ln for all n with F = , F = , L = , L = . Some papers related to Fibonacci numbers and Lucas numbers can also be found in []. From our results we can also deduce the following identities.

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

[summationdisplay]

a+b+c=n

Fm(a+) Fm(b+) Fm(c+)

= (n + )FmFm

Fm Fm(n+)

(n + )(n + )Fm (n + )(n + )Fm

Fm Fm(n+).

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

[summationdisplay]

a+b+c+d=n

Fm(a+) Fm(b+) Fm(c+) Fm(d+)

= (n + )Fm((n + n + )Lm (n + n + ))

Fm Fm(n+)

FmFm(n + )((n + n + )Lm (n + n ))

Fm Fm(n+).

Taking m = in Corollaries and we may immediately deduce the following.

Corollary For any nonnegative integer n, we have the identities

[summationdisplay]

a+b+c=n

F(a+) F(b+) F(c+) =

(n + )

F(n+) +

(n + )(n )

F(n+)

and

[summationdisplay]

a+b+c+d=n

F(a+) F(b+) F(c+) F(d+)

= (n + )(n + n + )

F(n+)

(n + )(n + n + )

F(n+).

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.

Siyi Advances in Dierence Equations (2015) 2015:355 Page 5 of 8

Lemma For any positive integers n k > , we have the identity

U(k)n(x) = (k )x

x U(k)n(x) +

dydx + n(n + )y = (n = , , , . . .).

So for any positive integer n k > , we have

x[parenrightbig]U

n (x) = xU n(x) n(n + )Un(x). ()

Dierentiating () repeatedly (k ) times we obtain

x[parenrightbig]U(k)

n (x) (k )xU(k)n(x) (k )(k )U(k)n(x)

= xU(k)n(x) + (k )U(k)(x) n(n + )U(k)n(x)

or

U(k)n(x) = (k )x

x U(k)n(x) +

(k )k n(n + ) x U(k)n(x).

Proof It is clear that the second kind Chebyshev polynomials Un(x) satisfy the dierential equation

x[parenrightbig] dydx x

(k )k n(n + )

x U(k)n(x).

This proves Lemma .

Lemma For any positive integers n k , we have the identity

U(k)n(x) =

( x)k [bracketleftbig]R(n, k, x) Un(x) + S(n, k, x) Un(x)[bracketrightbig],

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

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

x[parenrightbig]U n(x) = (n + )Un(x) nxUn(x)

or

U n(x) =

( x) [bracketleftbig](n + )Un(x) nxUn(x)[bracketrightbig]. ()

So Lemma holds for k = .

Assume that Lemma holds for all positive integers k m. That is, for all positive integers k m, we have

U(k)n(x) =

( x)k [bracketleftbig]Rk(n, k, x) Un(x) + Sk(n, k, x) Un(x)[bracketrightbig]. ()

Siyi Advances in Dierence Equations (2015) 2015:355 Page 6 of 8

Then for k = m + , from (), (), and Lemma we have

U(m+)n(x) = (m + )x

x U(m)n(x) +

(m )(m + ) n(n + ) x U(m)n(x)

=

m+(n, m + , x) Un(x) + Sm+(n, m + , x) Un(x)[bracketrightbig].

This proves Lemma by complete induction.

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

Tn[parenleftbig]Tm(x)[parenrightbig] = Tmn(x) and Un[parenleftbig]Tm(x)[parenrightbig] = Um(n

( x)m+ [bracketleftbig]R

+)(x)

Um(x) .

Proof See Lemma in Zhang [].

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 () and Lemma . In fact, substituting n by n + k in Lemma we have

U(k)n+k(x) = (k )x

x U(k)n+k(x) +

(k )k (n + k)(n + k + ) x U(k)n+k(x). ()

Combining identities () and () we may immediately deduce

[summationdisplay]

a+a++ak+=n

Ua(x) Ua(x) Uak+(x)

=

k k!

U(k)n+k(x) =

k k!

(k )x

x U(k)n+k(x)

n + n(k + ) + k

x U(k)n+k(x)

[bracketrightbigg].

This proves our theorem.

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

Now we prove Corollary . Taking k = in our theorem and noting that ( x)U n(x) =

(n + )Un(x) nxUn(x) we have

[summationdisplay]

a+b+c=n

Ua(x) Ub(x) Uc(x)

=

!

U n+(x) =

[bracketleftbigg] x

x U n+(x)

n + n +

x Un+(x)

[bracketrightbigg]

= x

( x)

[bracketleftbigg] n +

x Un+(x)

(n + )x

x Un+

[bracketrightbigg] (n + )(n + )

( x) Un+(x)

= (n + )x

( x) Un+(x)

(n + )(n + ) (n + )(n + )x

( x) Un+(x).

This proves Corollary .

To prove Corollary , taking k = in our theorem we have

[summationdisplay]

a+b+c+d=n

Ua(x) Ub(x) Uc(x) Ud(x)

=

!

U()n+(x) =

[bracketleftbigg] x

x U n+(x)

n + n +

x U n+(x)

[bracketrightbigg]

Siyi Advances in Dierence Equations (2015) 2015:355 Page 7 of 8

= x

( x)

[bracketleftbigg] x

x U n+(x)

(n + )(n + ) x Un+(x)

[bracketrightbigg] (n + )(n + )

( x) U n+(x)

= x (n + )(n + )( x)

( x)

[bracketleftbigg] n +

x Un+(x)

(n + )x

x Un+(x)

[bracketrightbigg]

(n + )(n + )x

( x) Un+(x)

= (n + )((n + n + )x (n + n + ))

( x) Un+(x)

x(n + )((n + n + )x (n + n ))

( x) Un+(x).

This proves Corollary .Now we prove Corollary . Taking x = in () and (), we note the identities

Tn[parenleftbigg]

[parenrightbigg] =

[bracketleftbigg][parenleftbigg]

+

[radicalbigg]

n

+ [parenleftbigg]

[radicalbigg]

n

[bracketrightbigg]

n

+ [parenleftbigg]

=

[bracketleftbigg][parenleftbigg] +

n

[bracketrightbigg] =

Ln ()

and

Un[parenleftbigg]

[parenrightbigg] =

[radicalBig]

[bracketleftbigg][parenleftbigg]

+

[radicalbigg]

n+

[parenleftbigg]

[radicalbigg]

n+

[bracketrightbigg] = Fn+. ()

Applying Lemma and () we also have

Un[parenleftbigg]Tm[parenleftbigg]

[parenrightbigg][parenrightbigg] = Um(n

+)()

Um()

Fm . ()

Taking x = Tm() in Corollary , applying (), (), and () we have

[summationdisplay]

a+b+c=n

Ua[parenleftbigg]Tm[parenleftbigg]

= Fm(n+)

[parenrightbigg][parenrightbigg]

Ub

[parenleftbigg]Tm[parenleftbigg]

[parenrightbigg][parenrightbigg]

Uc

[parenrightbigg][parenrightbigg] [parenleftbigg]Tm[parenleftbigg]

= [summationdisplay]

a+b+c=n

Fm(a+)

Fm

Fm(b+)

Fm

Fm(c+)

Fm

= (n + )Lm

( Lm)

Fm(n+)

Fm

(n + )(n + ) (n + )(n + )Lm

( Lm)

Fm(n+)

Fm

or

[summationdisplay]

a+b+c=n

Fm(a+) Fm(b+) Fm(c+)

= (n + )FmFm

Fm Fm(n+)

(n + )(n + )Fm (n + )(n + )Fm

Fm Fm(n+),

where we have used the identities Fm Lm = Fm and Lm = Fm.

Siyi Advances in Dierence Equations (2015) 2015:355 Page 8 of 8

Similarly, taking x = Tm() in Corollary , from (), (), and () we can also deduce the identity

[summationdisplay]

a+b+c+d=n

Fm(a+) Fm(b+) Fm(c+) Fm(d+)

= (n + )Fm((n + n + )Lm (n + n + ))

Fm Fm(n+)

FmFm(n + )((n + n + )Lm (n + n ))

Fm Fm(n+).

This proves Corollaries and .

Corollary follows from Corollary with m = , L = , F = F = , F = . This completes the proofs of our all results.

Competing interests

The author declares that they have no competing interests.

Authors contributions

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

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.

Received: 27 September 2015 Accepted: 7 November 2015

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