The Power Sums Involving Fibonacci Polynomials and Their Applications

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.

If x = 1, we have that {F n (x)} turns into Fibonacci sequences {F n }, and {L n (x)} turns into Lucas sequences {L n }.If x = 2, then F n (2) = P n , the nth Pell numbers, they are defined by P 0 = 0, P 1 = 1 and P n+2 = 2P n+1 + P n for all n ≥ 0. In fact, {F n (x)} is a second-order linear recursive polynomial, when x takes a different value x 0 , then F n (x 0 ) can become a different sequence.
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 [5], R. S. Melham proposed two interesting conjectures as follows: Conjecture 1.If m ≥ 1 is a positive integer, then the summation can be written as (F 2n+1 − 1) 2 P 2m−1 (F 2n+1 ), where P 2m−1 (x) is an integer coefficients polynomial with degree 2m − 1.
Sun et al. [7] solved Conjecture 1 completely.In fact, Ozeki [3] and Prodinger [4] 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 [7] 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 x = 1 and x = 2 in Theorem 1, we can instantly infer the two corollaries: Corollary 1.Let F n and L n be Fibonacci numbers and Lucas numbers, respectively.Then, for any positive integers n and h, we have the congruence Corollary 2. Let P n be nth Pell numbers.Then, for any positive integers n and h, we have the congruence It is clear that our Corollary 1 gave a new proof for Conjecture 1.

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 x 2 + 4 and F 2h+1 (x) − 1 are said to be relatively prime.
Proof.From the definition of F n (x) and binomial theorem, we have Thus, from Equation (1), we have the polynomial congruence Since x 2 + 4 is an irreducible polynomial of x, and (2h + 1)(−1) h − 1 is not divisible by (x 2 + 4) for all integer h ≥ 1, so, from (2), we can deduce that Lemma 1 is proved.Lemma 2. Let h and n be non-negative integers with h ≥ 1; then, we have Proof.We use mathematical induction to calculate the polynomial congruence for n.
, so we obtain the congruence which means that Lemma 2 is correct for n = 0 and 1. Assume Lemma 2 is right for all integers n = 0, 1, 2, • • • , k. Namely, where 0 ≤ n ≤ k.Thus, n = k + 1 ≥ 2, and we notice that 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 h ≥ 1; 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 n = 0, then it is obvious that ( 6) is correct.
Thus, n = 1 is fit for (6).Assume that ( 6) is correct for all integers n = 0, 1, 2, • • • , k. Namely, 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 [x] denotes the greatest integer ≤ x.
Proof.This formula due to Waring [15].It can also be found in Girard [14].