Abstract
The Girard and Waring formula and mathematical induction are used to study a problem involving the sums of powers of Fibonacci polynomials in this paper, and we give it interesting divisible properties. As an application of our result, we also prove a generalized conclusion proposed by R. S. Melham.
Keywords:
Fibonacci polynomials; Lucas polynomials; sums of powers; divisible properties; R. S. Melham’s conjectures MSC:
11B39
1. Introduction
For any integer , the famous Fibonacci polynomials and Lucas polynomials are defined as , , , and , for all . Now, if we let and , then it is easy to prove that
If , we have that turns into Fibonacci sequences , and turns into Lucas sequences . If , then , the nth Pell numbers, they are defined by , and for all . In fact, is a second-order linear recursive polynomial, when x takes a different value , then can become a different sequence.
Since the Fibonacci numbers and Lucas numbers occupy significant positions in combinatorial mathematics and elementary number theory, they are thus studied by plenty of researchers, and have gained a large number of vital conclusions; some of them can be found in References [,,,,,,,,,,,,,,]. For example, Yi Yuan and Zhang Wenpeng [] studied the properties of the Fibonacci polynomials, and proved some interesting identities involving Fibonacci numbers and Lucas numbers. Ma Rong and Zhang Wenpeng [] also studied the properties of the Chebyshev polynomials, and obtained some meaningful formulas about the Chebyshev polynomials and Fibonacci numbers. Kiyota Ozeki [] got some identity involving sums of powers of Fibonacci numbers. That is, he proved that
Helmut Prodinger [] extended the result of Kiyota Ozeki [].
In addition, regarding many orthogonal polynomials and famous sequences, Kim et al. have done a lot of important research work, obtaining a series of interesting identities. Interested readers can refer to References [,,,,,,]; we will not list them one by one.
In this paper, our main purpose is to care about the divisibility properties of the Fibonacci polynomials. This idea originated from R. S. Melham. In fact, in [], R. S. Melham proposed two interesting conjectures as follows:
Conjecture 1.
If is a positive integer, then the summation
can be written as , where is an integer coefficients polynomial with degree .
Conjecture 2.
If is an integer, then the summation
can be written as , where is an integer coefficients polynomial with degree .
Wang Tingting and Zhang Wenpeng [] solved Conjecture 2 completely. They also proved a weaker conclusion for Conjecture 1. That is,
can be expressed as , where is a polynomial of degree with integer coefficients.
Sun et al. [] solved Conjecture 1 completely. In fact, Ozeki [] and Prodinger [] indicated that the odd power sum of the first several consecutive Fibonacci numbers of even order is equivalent to the polynomial estimated at a Fibonacci number of odd order. Sun et al. in [] proved that this polynomial and its derivative both disappear at 1, and it can be an integer polynomial when a product of the first consecutive Lucas numbers of odd order multiplies it. This presents an affirmative answer to Conjecture 1 of Melham.
In this paper, we are going to use a new and different method to study this problem, and give a generalized conclusion. That is, we will use the Girard and Waring formula and mathematical induction to prove the conclusions in the following:
Theorem 1.
If n and h are positive integers, then we have the congruence
Taking and in Theorem 1, we can instantly infer the two corollaries:
Corollary 1.
Let and be Fibonacci numbers and Lucas numbers, respectively. Then, for any positive integers n and h, we have the congruence
Corollary 2.
Let be nth Pell numbers. Then, for any positive integers n and h, we have the congruence
where is called nth Pell–Lucas numbers.
It is clear that our Corollary 1 gave a new proof for Conjecture 1.
2. Several Lemmas
In this part, we will give four simple lemmas, which are essential to prove our main results.
Lemma 1.
Let h be any positive integer; then, we have
where and are said to be relatively prime.
Proof.
From the definition of and binomial theorem, we have
Since is an irreducible polynomial of x, and is not divisible by for all integer , so, from (2), we can deduce that
Lemma 1 is proved. □
Lemma 2.
Let h and n be non-negative integers with ; then, we have
Proof.
We use mathematical induction to calculate the polynomial congruence for n. Noting , , . Thus, if , then
If , then . Note that the identity , so we obtain the congruence
which means that Lemma 2 is correct for and 1.
Assume Lemma 2 is right for all integers . Namely,
where .
Thus, , and we notice that
and
From inductive assumption (3), we have
Now, we have achieved the results of Lemma 2. □
Lemma 3.
Let h and n be non-negative integers with ; then, we have the polynomial congruence
Proof.
For positive integer n, first note that , ,
and
Thus, from Labels (4) and (5), we know that, to prove Lemma 3, now we need to obtain the polynomial congruence
Now, we prove (6) by mathematical induction. If , then it is obvious that (6) is correct. If , we notice that , and we have
Thus, is fit for (6). Assume that (6) is correct for all integers . Namely,
for all .
Where , we notice
and
From inductive assumption (7) and Lemma 2, we have
Now, we attain Lemma 3 by mathematical induction. □
Lemma 4.
For all non-negative integers u and real numbers X, Y, we have the identity
in which denotes the greatest integer .
Proof.
This formula due to Waring []. It can also be found in Girard []. □
3. Proof of the Theorem
We will achieve the theorem by these lemmas. Taking , and in Lemma 4, we notice that , from the expression of
For any integer , from (8), we get
If , then, from (9), we can get
From Lemma 1, we know that , so, applying Lemma 3 and (10), we deduce that
This means that Theorem 1 is suitable for .
Assume that Theorem 1 is correct for all integers . Then,
for all integers .
When , from (9), we obtain
From Lemma 3, we have
Applying inductive hypothesis (12), we obtain
Combining (13), (14), (15) and Lemma 3, we have the conclusion
Note that , so (16) indicates the conclusion
Now, we apply mathematical induction to achieve Theorem 1.
Author Contributions
Conceptualization, L.C.; methodology, L.C and X.W.; validation, L.C. and X.W.; formal analysis, L.C.; investigation, X.W.; resources, L.C.; writing—original draft preparation, L.C.; writing—review and editing, X.W.; visualization, L.C.; supervision, L.C.; project administration, X.W.; all authors have read and approved the final manuscript.
Funding
This work is supported by the N. S. F. (11771351) and (11826205) of P. R. China.
Acknowledgments
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
Conflicts of Interest
The authors state that there are no conflicts of interest concerning the publication of this paper.
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