Some identities for the generalized Fibonacci polynomials by the Q(x) matrix

In this note, we obtain some identities for the generalized Fibonacci polynomial by using the Q(x) matrix. These identities including the Cassini identity and Honsberger formula can be applied to some polynomial sequences, such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials, Fermat polynomials, Fermat-Lucas polynomials, and so on.


Introduction
A second order polynomial sequence F n (x) is said to be the Fibonacci polynomial if for n ≥ 2 and x ∈ R, F n (x) = xF n−1 (x) + F n−2 (x) with F 0 (x) = 0 and F 1 (x) = 1. The Fibonacci polynomial and other polynomials attracted a lot of attention over the last several decades (see, for instance, [3,4,7,8,9,12]). Recently, the generalized Fibonacci polynomial is introduced and studied intensely by many authors [1,2,5,6], which is a generalization of the Fibonacci polynomial. Indeed, a polynomial sequence G n (x) in [5,6] is called the generalized Fibonacci polynomial if for n ≥ 2, G n (x) = c(x)G n−1 (x) + d(x)G n−2 (x) with initial conditions G 0 (x) and G 1 (x), where c(x) and d(x) are fixed non-zero polynomials in Q [x]. It should be noted that there is no unique generalization of Fibonacci polynomials. Following the similar definitions in [6], in this note, F n (x) is said to be the Fibonacci type polynomial if for n ≥ 2, F 0 (x) = 0, F 1 (x) = a and F n (x) = c(x)F n−1 (x) + d(x)F n−2 (x) where a ∈ R \ {0}. If for n ≥ 2, L 0 (x) = q, L 1 (x) = b(x) and L n (x) = c(x)L n−1 (x) + d(x)L n−2 (x), then the polynomial sequence L n (x) is called the Lucas type polynomial, where q ∈ R \ {0} and b(x) is a fixed non-zero polynomial in Q [x]. Naturally, both F n (x) and L n (x) are the generalized Fibonacci polynomials. We note that if we assume F 1 (x) = a = 1, then F n (x) is the Fibonacci type polynomial given in [6]. In addition, the definition of L n (x) is the same with that of Flórez et al [6] if |q| = 1 or 2, and c(x) = 2 q b(x). In other words, our definitions of F n (x) and L n (x) are generalizations of those in [6].
Since the investigation of identities for polynomial sequences F n (x) and L n (x) received less attention than their numerical sequences, Flórez et al [6] collected and proved many identities for both F n (x) and L n (x) by applying their Binet formulas mostly, when certain special initial conditions were satisfied for F n (x) and L n (x). These identities can be applied to Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials, Fermat polynomials, Fermat-Lucas polynomials, Chebyshev first kind polynomials, Chebyshev second kind polynomials, Jacobsthal polynomials, Jacobsthal-Lucas polynomials, and Morgan-Voyce polynomials. Indeed, all polynomial sequences in the upper part of Table 1 below are the Fibonacci type polynomials. On the other hand, those in the lower part of Table 1 are the Lucas type polynomials. Table 1 is the rearrangement of [6, Table  1]. Table 1.

Polynomial
Initial value Initial value Recursive Formula In the note, by using the so called the Q(x) matrix of Fibonacci type polynomials rather than the Binet formulas, we will obtain some new identities or recover some well-known ones including the Cassini identity and Honsberger formula for F n (x) and L n (x). In Section 2, we will present the results for the Fibonacci type polynomial F n (x). Relying on Section 2, the identities of the Lucas type polynomial L n (x) will be demonstrated in Section 3.

Fibonacci type polynomials
In this section, we will provide and prove some identities for the Fibonacci type polynomial F n (x) by applying the Fibonacci type Q(x) matrix. The original Fibonacci Q matrix was introduced by Charles H. King in his master thesis (cf. [10]), and given by The Fibonacci Q matrix is connected to the Fibonacci sequence F n , which is defined as below Indeed, it is noted in [7] that Using this relation above, some familiar identities can be obtained. For instance, det F n+1 F n F n F n−1 = det 1 1 1 0 n implies the Cassini identity F n+1 F n−1 − F 2 n = (−1) n . Also, using this equality Q n+m = Q n Q m , one can deduce the Honsberger formula.
In the following, we will apply some similar idea of Q matrix from the numerical cases [11] to the Fibonacci type polynomials. For n ≥ 2 and x ∈ R, the Fibonacci type polynomial F n (x) is defined by Here we define the Fibonacci type Q(x) matrix by We note that if F n (x) = P n (x) is the Pell polynomial as defined in Table 1, then which appeared in [9]. In addition, we observe that On the other hand, Hence we have the following result.
Proof. Let n = 1. Then Assume the equality holds for n = k. Then we have By induction, the result follows.
The Cassini identity of the Fibonacci type polynomial F n (x) can be obtained below by Theorem 2.1.
Corollary 2.2. Let F n (x) be the Fibonacci type polynomial. Then for each n ∈ N, Proof. By Theorem 2.1, we have . By Corollary 2.2, we recover the Cassini identity in [4], Example 2.4. Let F n (x) be the Pell polynomial P n (x) as defined in Table 1. By Corollary 2.2, is the Jacobsthal polynomial as defined in Table 1. By Corollary 2.2, one can obtain the Cassini identity for the Jacobsthal polynomial below By Corollary 2.2, we have the result below.
Corollary 2.6. Let F n (x) be the Fibonacci type polynomial. Then for each n ∈ N, Proof. By By applying Q n+m (x) = Q n (x)Q m (x), we give the Honsberger's formula for the the Fibonacci type polynomials below. Corollary 2.7. Let F n (x) be the Fibonacci type polynomial. Then for each n, m ∈ N, .
Using Q n−m (x) = Q n (x)Q −m (x) for n ≥ m, we next will prove the d'Ocagne identity for F n (x). Here we need to assume d(x) = 0 for each x ∈ R so that Q(x) is invertible. Moreover, note that Corollary 2.11. Let F n (x) be the Fibonacci type polynomial, and let d(x) = 0 for each x ∈ R. Then for each n, m ∈ N with n ≥ m, .
Hence considering the (1, 2) entry of the first matrix in the equality above, Example 2.12. Let F n (x) be the Fibonacci polynomial F n (x) as defined in Table  1. By Corollary 2.11, which is the d'Ocagne identity in [4,Corollary 8], and the identity (47) of [6, Proposition 3].
We note that Using this equality, one can obtain the following expression of F n (x).
Theorem 2.13. Let F n (x) be the Fibonacci type polynomial. Then for each n, p ∈ N, .

Lucas type polynomials
Based on the results of Fibonacci type polynomials, some identities of Lucas type polynomials will be demonstrated in this section. Throughout this section, we assume L n (x) and F n (x) have the same recursive formula with L 0 (x) = F 1 (x), that is, for n ≥ 2, where a ∈ R \ {0}. By applying Theorem 2.1, one can connect L n (x) with F n (x) below.
Theorem 3.1. Let F n (x) and L n (x) be the Fibonacci type polynomial and Lucas type polynomial respectively with L 0 (x) = F 1 (x) = a. Then for each n ∈ N, Proof. First, we will prove L n (x) = b(x) a F n (x) + d(x)F n−1 (x) holds for each n ∈ N. Let n = 1. Then Let n = 2. Then Assume this equality hods for n = k − 1 and k. Let n = k + 1. Then On the other hand, we have One has the result by these two equalities Next, we will demonstrate the relation between Lucas type polynomials and the Fibonacci type Q(x) matrix .
Theorem 3.2. Let L n (x) be the Lucas type polynomial. Then for each n ∈ N, Proof. By Theorem 2.1 and Theorem 3.1, we have Using Theorem 3.2, one has the Cassini identity for the Lucas type polynomial L n (x).
Proof. By Theorem 3.2, we have Example 3.4. Let a = 2, b(x) = 2x, c(x) = 2x, d(x) = 1 in Eq. (2). Then L n (x) = D n (x) is the Pell-Lucas polynomial as defined in Table 1. By Corollary 3.3, the Cassini identity for the Pell-Lucas polynomial D n (x) is given by By Corollary 3.3, we have the result below.
Corollary 3.5. Let L n (x) be the Lucas type polynomial. Then for each n ∈ N, Proof. By we have Using Q 2 (x) = c(x)Q(x) + d(x)I again, we have the expression of L n (x).
Theorem 3.6. Let L n (x) be the Lucas type polynomial. Then for each n, p ∈ N, Proof. By Theorem 3.2, we have By considering the (2, 2) entry of the first matrix in the above equality, we have Example 3.7. Let L n (x) be the Morgan-Voyce polynomial C n (x) in which a = 2, b(x) = x + 2, c(x) = x + 2, d(x) = −1 in Eq. (2). By Theorem 3.6, we have Finally, we end up this note by providing an identity in which F n (x) and L n (x) are involved.
Proposition 3.8. Let F n (x) and L n (x) be the Fibonacci type polynomial and Lucas type polynomial respectively with L 0 (x) = F 1 (x) = a. Then for each n, m ∈ N, .
Then by the (2, 2) entry of the first matrix in the above equality, we have for each n, m ∈ N.
Example 3.9. Let F n (x) and L n (x) be the Jacobsthal polynomial J n (x) and the Jacobsthal-Lucas polynomial Λ n (x) respectively, as defined in Table 1. Then Λ 0 (x) = J 1 (x) = 1 which satisfies the condition in Proposition 3.8. Hence we have the following equality for J n (x) and Λ n (x): Proof. By Theorem 3.2 and Q n−m (x) = Q n (x)Q −m (x), we have .
Then considering the (2, 2) entry of the first matrix in the above equality, we have a(−d(x)) m L n−m (x) = L n (x)F m+1 (x) − L n+1 (x)F m (x).