- Research
- Open access
- Published:
On the Lucas polynomials and some of their new identities
Advances in Difference Equations volume 2018, Article number: 126 (2018)
Abstract
The main purpose of this paper is, using the elementary and combination methods, to study the arithmetical properties of the Lucas polynomials and to obtain some new and interesting identities for them.
1 Introduction
For any non-negative integer n, the Fibonacci polynomials \(\{F_{n}(x)\}\) and Lucas polynomials \(\{L_{n}(x)\}\) are defined by the second order linear recursive formulas \(F_{n+2}(x)=xF_{n+1}(x)+F_{n}(x)\) and \(L_{n+2}(x)=xL_{n+1}(x)+L_{n}(x)\) with \(F_{0}(x)=0\), \(F_{1}(x)=1\), \(L_{0}(x)=2\), and \(L_{1}(x)=x\). The general terms of \(F_{n}(x)\) and \(L_{n}(x)\) are given by
and
where \(\binom{m}{n}=\frac{m!}{n!(m-n)!}\), and \([x]\) denotes the greatest integer ≤x.
It is easy to prove the identities
and
If \(x=1\), then \(\{F_{n}(x)\}\) becomes the famous Fibonacci sequences \(\{F_{n}\}\) and \(\{L_{n}(x)\}\) becomes the Lucas sequences \(\{L_{n}\}\).
These sequences and polynomials occupy very important positions in the theory and application of mathematics, so many scholars have studied their various arithmetical properties and obtained a series of important results. For example, Ozeki [1] proved the identity
Prodinger [2] studied the more general summation \(\sum_{k=0}^{n}F_{2k+ \delta }^{2m+1+\epsilon }\), where \(\delta , \epsilon \in \{0, 1\}\), and obtained many interesting results.
Ma and Zhang [3] used the properties of Chebyshev polynomials to obtain some identities involving Fibonacci numbers and Lucas numbers. Wang and Zhang [4] proved some divisible properties involving Fibonacci numbers and Lucas numbers. Some of other related papers can also be found in references [5–15], here we are not going to list them all.
In this paper, we shall use the elementary and combination methods to study the arithmetical properties of Lucas polynomials, and give some new identities for them. That is, we shall prove the following results.
Theorem 1
For any positive integer h and integer \(k \geq 0\), we have
Theorem 2
For any integers h and \(k\geq 0\), we have
Theorem 3
For any integers \(n\geq 1\) and \(h\geq 0\), we have the identity
Theorem 4
For any integers \(n\geq 1\) and \(h\geq 0\), we have the identity
Taking \(k=0\), from Theorems 1 and 2 we can deduce the following:
Corollary 1
For any positive integer h, we have the identities
and
If \(x=1\) and \(k=0\), then we also have the following:
Corollary 2
For any positive integer h, we have the identities
From Theorems 3 and 4 we can deduce the following corollaries.
Corollary 3
For any integers \(n\geq 1\) and \(h\geq 0\), we have
Corollary 4
For any integers \(n\geq 1\) and \(h\geq 0\), we have
2 Several simple lemmas
Lemma 1
For any positive integers n, we have the identities
Proof
Note the identities \(( x+\sqrt{x^{2}+4} ) '= 1+\frac{x}{\sqrt{x ^{2}+4}}=\frac{x+\sqrt{x^{2}+4}}{\sqrt{x^{2}+4}}\) and \(( x-\sqrt{x ^{2}+4} ) '= 1-\frac{x}{\sqrt{x^{2}+4}}=-\frac{x- \sqrt{x^{2}+4}}{\sqrt{x^{2}+4}}\). From the definitions of the polynomials \(F_{n}(x)\) and \(L_{n}(x)\), we have
Applying (3), the integration by parts, and the recursive formulae of \(L_{n}(x)\) and \(F_{n}(x)\), we have
If \(n=2k\), then note that \(L_{2k+1}(0)=L_{2k-1}(0)=0\). From (4) we have
If \(n=2k+1\), then by (2) we have \(L_{2k+2}(0)=L_{2k}(0)=2\). From (4) we have
Now Lemma 1 follows from (5) and (6). □
Lemma 2
For any positive integer n and non-negative integer k, we have the identity
Proof
Let \(\alpha =\frac{x+\sqrt{x^{2}+4}}{2}\) and \(\beta = \frac{x-\sqrt{x^{2}+4}}{2}\). Then replace x by \(L_{2k+1}(x)\) in (2) and note that \(\alpha^{2k+1}\beta^{2k+1}=-1\), we have
and
From (2) we have the identity
This proves Lemma 2. □
Lemma 3
For any non-negative integer n, we have the identities
and
Proof
From the definition of \(L_{n}(x)\) we know that \(L_{2k}(x)\) is an even function. So we may suppose that
Taking \(x=2i\cos \theta \) in (7) and noting that \(x^{2}+4=4-4\cos^{2} \theta =4\sin^{2}\theta \), from Euler’s formula we have
Note the identities
and
or
Combining (7), (9), and (11), we can deduce the first identity of Lemma 3.
Similarly, since \(L_{2k+1}(x)\) is an odd function, we can suppose that
Taking \(x=2i\cos \theta \) in (12) and noting that
we have
From (10) and (13) we may immediately deduce that
Now the second identity of Lemma 3 follows from (12) and (14). □
3 Proofs of the theorems
Using the lemmas in Sect. 2, we can prove our theorems easily. First we prove Theorem 2. Similarly, we can also deduce Theorem 1 and then omit its proving process here. From Lemma 1 and the definition of \(L_{n}(x)\), we have
Applying (15) and Lemma 1, we have
or
Now Theorem 2 follows from (16) and Lemma 2 with \(x=L_{2k+1}(y)\).
To prove Theorem 3, taking \(x=L_{2h+1}(y)\) in Lemma 3, we have
Integrating for y from 0 to x in (17), then applying Lemma 1, we may immediately deduce
This proves Theorem 3.
Similarly, we can also deduce Theorem 4.
References
Ozeki, K.: On Melham’s sum. Fibonacci Q. 46/47, 107–110 (2008/2009)
Prodinger, H.: On a sum of Melham and its variants. Fibonacci Q. 46/47, 207–215 (2008/2009)
Ma, R., Zhang, W.: Several identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 45, 164–170 (2007)
Wang, T., Zhang, W.: Some identities involving Fibonacci. Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roum. 55, 95–103 (2012)
Melham, R.S.: Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers. Fibonacci Q. 46/47, 312–315 (2008/2009)
Li, X.: Some identities involving Chebyshev polynomials. Math. Probl. Eng. 2015, Article ID 950695 (2015)
Ma, Y., Lv, X.: Several identities involving the reciprocal sums of Chebyshev polynomials. Math. Probl. Eng. 2017, Article ID 4194579 (2017)
Kim, D.S., Dolgy, D.V., Kim, T., Rim, S.H.: Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials. Proc. Jangjeon Math. Soc. 15, 361–370 (2012)
Kim, D.S., Kim, T., Lee, S.: Some identities for Bernoulli polynomials involving Chebyshev polynomials. J. Comput. Anal. Appl. 16, 172–180 (2014)
Kim, T., Kim, D.S., Seo, J.J., Dolgy, D.V.: Some identities of Chebyshev polynomials arising from non-linear differential equations. J. Comput. Anal. Appl. 23, 820–832 (2017)
He, Y.: Some new results on products of Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 431, 34–46 (2015)
He, Y., Wang, C.P.: New symmetric identities involving the Eulerian polynomials. J. Comput. Anal. Appl. 17, 498–504 (2014)
He, Y., Kim, D.S.: General convolution identities for Apostol–Bernoulli, Euler and Genocchi polynomials. J. Nonlinear Sci. Appl. 9, 4780–4797 (2016)
Chen, L., Zhang, W.: Chebyshev polynomials and their some interesting applications. Adv. Differ. Equ. 2017, 303 (2017)
Yi, Y., Zhang, W.: Some identities involving the Fibonacci polynomials. Fibonacci Q. 40, 314–318 (2002)
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 N.S.F. (Grant No. 11771351) of P.R. China.
Author information
Authors and Affiliations
Contributions
All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that she has no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access 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.
About this article
Cite this article
Jin, Z. On the Lucas polynomials and some of their new identities. Adv Differ Equ 2018, 126 (2018). https://doi.org/10.1186/s13662-018-1527-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-018-1527-9