Li and Wenpeng Advances in Difference Equations (2017) 2017:303 DOI 10.1186/s13662-017-1365-1
RESEARCH
Open Access
Chebyshev polynomials and their some interesting applications Chen Li and Zhang Wenpeng* *
Correspondence:
[email protected] School of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China
Abstract The main purpose of this paper is by using the definitions and properties of Chebyshev polynomials to study the power sum problems involving Fibonacci polynomials and Lucas polynomials and to obtain some interesting divisible properties. MSC: Primary 11B39 Keywords: Chebyshev polynomials; Fibonacci polynomials; Lucas polynomials; power sums; divisible property
1 Introduction For any integer n ≥ , the first kind Chebyshev polynomials {Tn (x)} and the second kind Chebyshev polynomials {Un (x)} are defined by T (x) = , T (x) = x, U (x) = , U (x) = x and Tn+ (x) = xTn+ (x) – Tn (x), Un+ (x) = xUn+ (x) – Un (x) for all n ≥ . If we write α = √ √ α(x) = x + x – and β = β(x) = x – x – for the sake of simplicity, then we have [n]
n (n – k – )! Tn (x) = α n + β n = (x)n–k , (–)k k!(n – k)!
|x| <
()
k=
and [n]
(n – k)! n+ (–)k α – β n+ = (x)n–k , Un (x) = α–β k!(n – k)!
|x| < .
()
k=
Fibonacci polynomials {Fn (x)} and Lucas polynomials {Ln (x)} are defined by F (x) = , Fn+ (x) = xFn+ (x) + Fn (x), Ln+ (x) = xLn+ (x) + Ln (x) for F (x) = , L (x) = , L (x) = x and √ √ x+ x + x– x + all n ≥ . If we write U(x) = and V (x) = , then we have Fn (x) =
n U (x) – V n (x) and U(x) – V (x)
Ln (x) = U n (x) + V n (x) for all n ≥ .
These polynomials occupy a very important position in the theory and application of mathematics, so many scholars have studied their various properties and obtained a series of interesting and important results. See references [–] for Chebyshev polynomials and © The Author(s) 2017. 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.
Li and Wenpeng Advances in Difference Equations (2017) 2017:303
[–] for Fibonacci and Lucas polynomials. For example, Li Xiaoxue [] proved some identities involving power sums of Tn (x) and Un (x). As some applications of these results, she obtained some divisible properties involving Chebyshev polynomials. Precisely, she proved the congruence
U (x)U (x)U (x) · · · Un (x) ·
h
n+ Tm (x) ≡ mod Uh (x) – .
m=
In this paper, we shall use the definition and properties of Chebyshev polynomials to study the power sum problem involving Fibonacci and Lucas polynomials and prove some new divisible properties involving these polynomials. That is, we shall prove the following two generalized conclusions. Theorem Let n and h be non-negative integers with h ≥ . Then, for any odd number l ≥ , we have the congruence Ll (x)Ll (x) · · · Ll(n+) (x) ·
h
Ln+ ml (x) ≡ mod Ll(h+) (x) + Ll (x) .
m=
Theorem Let n and h be non-negative integers with h ≥ . Then, for any even number l ≥ , we have the congruence Fl (x)Fl (x) · · · Fl(n+) (x) ·
h
Ln+ ml (x) ≡ mod Fl(h+) (x) + Fl (x) .
m=
Especially for l = and , from our theorems we may immediately deduce the following two corollaries. Corollary For any non-negative integers n and h with h ≥ , we have L (x)L (x) · · · Ln+ (x) ·
h
Ln+ m (x) ≡ mod Lh+ (x) + x .
m=
Corollary For any non-negative integers n and h with h ≥ , we have F (x)F (x) · · · F(n+) (x) ·
h
Ln+ m (x) ≡ mod F(h+) (x) + x .
m=
Some notes: In our theorems, the range of the summation for m is from to h. If the range of the summation is ≤ m ≤ h, then it is very easy to prove the following corresponding results: For any odd number l ≥ , we have the polynomial congruence Ll (x)Ll (x) · · · Ll(n+) (x) ·
h m=
Ln+ ml (x) ≡ mod Ll(h+) (x) – Ll (x) .
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For any even number l ≥ , we have the polynomial congruence Fl (x)Fl (x) · · · Fl(n+) (x) ·
h
Ln+ ml (x) ≡ mod Fl(h+) (x) – Fl (x) .
m=
Taking l = and , and noting that L (x) = F (x) = x, then from these two congruences we can deduce the following: L (x)L (x) · · · Ln+ (x) ·
h
Ln+ m (x) ≡ mod Lh+ (x) – x
m=
and F (x)F (x) · · · F(n+) (x) ·
h
Ln+ m (x) ≡ mod F(h+) (x) – x .
m=
Therefore, our theorems are actually an extension of references [] and [].
2 Several simple lemmas To complete the proofs of our theorems, we need some new properties of Chebyshev polynomial, which we summarize as the following three lemmas. Lemma For any integers m, n ≥ , we have the identity Tn
Lm (x) = · Lmn (x). √
√
Proof Let α = x+ x + and β = x– x + , then Lm (x) = α m + β m , α · β = – and α m · β m = . Replace x by Lm (x) in Tn (x) and note that Lm (x) + L (x) – m = α m + β m + · α m + β m + – m = α + β m + α m – β m = α m , Lm (x) – L (x) – m = α m + β m – · α m + β m + – = α m + β m – α m – β m = β m . From the definition of Tn (x) Tn (x) =
√ n √ n
x+ x – + x– x – ,
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we have the identity
Tn Lm (x) n n Lm (x) + L (x) – + Lm (x) – L (x) – = m m
= α mn + β mn = · Lmn (x). This proves Lemma .
Lemma Let n and h be non-negative integers with h ≥ . Then, for any odd number l ≥ , we have the congruence Ll(h+)(n+) (x) + Ll(n+) (x) ≡ mod Ll(h+) (x) + Ll (x) . Proof We prove this polynomial congruence by complete induction for n ≥ . It is clear that Lemma is true for n = . If n = , then note that l and Ll(h+) (x) = Ll(h+) (x) – Ll(h+) (x), we have Ll(h+) (x) + Ll (x) = Ll(h+) (x) + Ll(h+) (x) + Ll (x) + Ll (x) = Ll(h+) (x) + Ll (x) Ll(h+) (x) + Ll(h+) (x)Ll (x) + Ll (x) + ≡ mod Ll(h+) (x) + Ll (x) . That is to say, Lemma is true for n = . Suppose that Lemma is true for all integers n = , , . . . , k. That is, Ll(h+)(n+) (x) + Ll(n+) (x) ≡ mod Ll(h+) (x) + Ll (x) for all ≤ n ≤ k. Then, for n = k + ≥ , note the identities Ll(h+) (x)Ll(h+)(n+) (x) = Ll(h+)(n+) (x) + Ll(h+)(n–) (x) and Ll(h+) (x) = Ll(h+) (x) + ≡ Ll (x) + mod Ll(h+) (x) + Ll (x) , applying inductive hypothesis (), we have Ll(h+)(n+) (x) + Ll(n+) (x) = Ll(h+)(k+) (x) + Ll(k+) (x)
()
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= Ll(h+) (x)Ll(h+)(k+) (x) – Ll(h+)(k–) (x) + Ll (x)Ll(k+) (x) – Ll(k–) (x) ≡ Ll (x) + Ll(h+)(k+) (x) – Ll(h+)(k–) (x) + Ll (x) + Ll(k+) (x) – Ll(k–) (x) ≡ Ll (x) + Ll(h+)(k+) (x) + Ll(k+) (x) – Ll(h+)(k–) (x) + Ll(k–) (x) ≡ Ll(h+) (x) + Ll (x) . Now Lemma follows from complete induction.
Lemma Let n and h be non-negative integers with h ≥ . Then, for any even number l ≥ , we have the congruence Fl(h+)(n+) (x) + Fl(n+) (x) ≡ mod Fl(h+) (x) + Fl (x) . Proof We can also prove Lemma by complete induction. If n = , then it is clear that Lemma is true. If n = , then note that | l and Fl(h+) (x) = (x + )Fl(h+) (x) + Fl(h+) (x), we have Fl(h+) (x) + Fl (x) = x + Fl(h+) (x) + Fl(h+) (x) + x + Fl (x) + Fl (x) (x) + Fl(h+) (x)Fl (x) + Fl (x) = x + Fl(h+) (x) + Fl (x) Fl(h+) + Fl(h+) (x) + Fl (x) ≡ mod Fl(h+) (x) + Fl (x) . So Lemma is true for n = . Suppose that Lemma is true for all integers n = , , . . . , k. That is, Fl(h+)(n+) (x) + Fl(n+) (x) ≡ mod Fl(h+) (x) + Fl (x) for all ≤ n ≤ k. Then, for n = k + , note the identities Ll(h+) (x)Fl(h+)(n+) (x) = Fl(h+)(n+) (x) + Fl(h+)(n–) (x) and Ll(h+) (x) = x + Fl(h+) (x) + ≡ x + Fl (x) + mod Fl(h+) (x) + Fl (x) , applying inductive hypothesis (), we have Fl(h+)(n+) (x) + Fl(n+) (x) = Fl(h+)(k+) (x) + Fl(k+) (x)
()
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= Ll(h+) (x)Fl(h+)(k+) (x) – Fl(h+)(k–) (x) + Ll (x)Fl(k+) (x) – Fl(k–) (x)
≡ x + Fl (x) + Fl(h+)(k+) (x) – Fl(h+)(k–) (x)
+ x + Fl (x) + Fl(k+) (x) – Fl(k–) (x)
≡ x + Fl (x) + Fl(h+)(k+) (x) + Fl(k+) (x) – Fl(h+)(k–) (x) + Fl(k–) (x) ≡ Fl(h+) (x) + Fl (x) . This completes the proof of Lemma .
3 Proofs of the theorems In this section, we shall prove our theorems by mathematical induction. Replace x by Lml (x) in (), from Lemma we have
h
Tn+
m=
Lml (x)
·
=
h Lml(n+) (x) · m=
=
n h (n – k)! n + · (–)k Ln+–k (x) k!(n + – k)! m= ml k=
or h
Lml(n+) (x) – (–)n (n + )Lml (x) m=
= (n + ) ·
n– (–)k k=
h (n – k)! Ln+–k (x). k!(n + – k)! m= ml
()
Note the identities h
Lml(n+) (x)
m=
=
h
α ml(n+) + β ml(n+)
m=
=
α (h+)l(n+) – β (h+)l(n+) – + α l(n+) – β l(n+) –
=
α l(h+)(n+) – (–)l β l(n+) β l(h+)(n+) – (–)l α l(n+) + α l(n+) – (–)l β l(n+) β l(n+) – (–)l α l(n+)
=
Ll(h+)(n+) (x) + Ll(n+) (x) Ll(n+) (x)
if l;
()
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and h
Lml(n+) (x) =
m=
Fl(h+)(n+) (x) + Fl(n+) (x) Fl(n+) (x)
if | l.
()
If l is an odd number, then from () and () we have Ll(h+) (x) + Ll (x) Ll(h+)(n+) (x) + Ll(n+) (x) – (–)n (n + ) Ll(n+) (x) Ll (x) = (n + ) ·
n– (–)k k=
h (n – k)! Ln+–k (x). k!(n + – k)! m= ml
()
Now we prove Theorem by mathematical induction. If n = , then from (), Lemma and note that l we have
Ll (x)Ll (x)
h
Lml (x)
m=
Ll(h+) (x) + Ll (x) Ll(h+) (x) + Ll (x) + Ll (x) Ll (x) ≡ mod Ll(h+) (x) + Ll (x) .
= Ll (x)Ll (x)
()
That is, Theorem is true for n = . Suppose that Theorem is true for all integers ≤ n ≤ s. Then, for n = s + , from () we have Ll(h+) (x) + Ll (x) Ll(h+)(s+) (x) + Ll(s+) (x) + (–)s (s + ) Ll(s+) (x) Ll (x) = (s + ) ·
s
(–)k
k=
=
h
h (s + – k)! s+–k L (x) k!(s + – k)! m= ml
Ls+ ml (x) + (s + ) ·
m=
s h (s + – k)! s+–k (–)k L (x). k!(s + – k)! m= ml
()
k=
From Lemma we have Ll(h+)(s+) (x) + Ll(s+) (x) Ll(s+) (x) ≡ mod Ll(h+) (x) + Ll (x) .
Ll (x)Ll (x) · · · Ll(s+) (x) ·
()
Applying inductive assumption, we have
Ll (x)Ll (x) · · · Ll(s+) (x)
s h (s + – k)! s+–k (–)k L (x) k!(s + – k)! m= ml k=
≡ mod Ll(h+) (x) + Ll (x) .
()
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Combining ()-() and Lemma , we can deduce the congruence Ll (x)Ll (x) · · · Ll(s+) (x) ·
h
Ls+ ml (x) ≡ mod Ll(h+) (x) + Ll (x) .
m=
This proves Theorem by mathematical induction. Now we prove Theorem . If | l, then from () and () we have Fl(h+) (x) + Fl (x) Fl(h+)(n+) (x) + Fl(n+) (x) – (–)n (n + ) Fl(n+) (x) Fl (x) = (n + ) ·
n– (–)k k=
h (n – k)! Ln+–k (x). k!(n + – k)! m= ml
()
Applying (), Lemma and the method of proving Theorem , we may immediately deduce the congruence Fl (x)Fl (x) · · · Fl(n+) (x) ·
h
Ln+ ml (x) ≡ mod Fl(h+) (x) + Fl (x) .
m=
This completes the proof of Theorem .
Acknowledgements The author would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the NSF Grant No. 11771351 of P.R. China. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 22 August 2017 Accepted: 18 September 2017 References 1. Li, X: Some identities involving Chebyshev polynomials. Math. Probl. Eng. 2015, Article ID 950695 (2015) 2. Ma, Y, Lv, X: Several identities involving the reciprocal sums of Chebyshev polynomials. Math. Probl. Eng. 2017, Article ID 4194579 (2017) 3. Clemente, C: Identities and generating functions on Chebyshev polynomials. Georgian Math. J. 19(3), 427-440 (2012) 4. Chan-Lye, L, Wong, KB: On Chebyshev’s polynomials and certain combinatorial identities. Bull. Malays. Math. Soc. 34(2), 279-286 (2011) 5. Doha, EH, Bhrawy, AH, Ezz-Eldien, SS: Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method. Cent. Eur. J. Phys. 11(10), 1494-1503 (2013) 6. Bhrawy, AH, Zaky, MA: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876-895 (2015) 7. Doha, EH, Bhrawy, AH, Hafez, RM, Abdelkawy, MA: A Chebyshev-Gauss-Radau scheme for nonlinear hyperbolic system of first order. Appl. Math. Inf. Sci. 8(2), 535-544 (2013) 8. Doha, EH, Bhrawy, AH: A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. Numer. Methods Partial Differ. Equ. 25(3), 712-739 (2009) 9. Wang, T, Zhang, H: Some identities involving the derivative of the first kind Chebyshev polynomials. Math. Probl. Eng. 2015, Article ID 146313 (2015) 10. Bircan, N, Pommerenke, C: On Chebyshev polynomials and GL(2, Z/pZ). Bull. Math. Soc. Sci. Math. Roum. 103, 353-364 (2012) 11. Kim, T, Kim, DS, Seo, JJ, Dolgy, DV: Some identities of Chebyshev polynomials arising from non-linear differential equations. J. Comput. Anal. Appl. 23(5), 820-832 (2017) 12. Kim, DS, Kim, T, Lee, S: Some identities for Bernoulli polynomials involving Chebyshev polynomials. J. Comput. Anal. Appl. 16(1), 172-180 (2014)
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