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Identities for the generalized Fibonacci polynomial

In this paper we generalize identities from Fibonacci numbers to the generalized Fibonacci polynomial (GFP). We classify the GFP into two types, namely Fibonacci type and Lucas type. A Fibonacci type polynomial is equivalent to a Lucas type polynomial if they both satisfy the same recurrence relations. Most of the identities provide relationships between two equivalent polynomials. In particular, each type of identities in this paper relate the following GFP: Fibonacci with Lucas, Pell with Pell-Lucas, Fermat with Fermat-Lucas, both types of Chebyshev polynomials, Jacobsthal with Jacobsthal-Lucas and both types of Morgan-Voyce.


Introduction
The Generalized Fibonacci Polynomial (GFP) is a natural generalization of the sequence F n (x) = xF n−1 (x) + 1F n−2 (x) where F 0 (x) = 0, F 1 (x) = 1 -called the Fibonacci polynomial sequence-. However, there is no unique generalization of this sequence, one can refer to articles by several authors like André-Jeannin [1,2], Bergum et al. [4], and Flórez et al. [5,6], to see this. In this paper we use the definition of the GFP introduced by Flórez, Higuita, and Mukherjee [5,6]. Flórez et al. [5], proved the strong divisibility property for the GFP, but while working on it the authors needed some identities involving the polynomials. When searching through the existing literature they realized that there were a limited number of those identities, even just for Fibonacci polynomials.
The study of identities for Fibonacci polynomials and Lucas polynomials have received less attention than their counterparts for numerical sequences, even if many of these identities can be proved easily. A natural question to ask is: under what conditions is it possible to extend identities that already exist for Fibonacci and Lucas numbers to the GFP? We observe here that the identities involving Fibonacci and Lucas numbers extend naturally to the GFP that satisfy closed formulas similar to the Binet formulas satisfied by Fibonacci and Lucas numbers.
While this paper does not intend to prove new numerical identities, we have concluded with a substantial list of known numerical identities extended to the GFP. We also adapt some known identities given for Fibonacci or Lucas polynomials to the GFP.
Bergum and Hoggatt [4] gave a list of more than twenty identities for their definition of generalized Fibonacci polynomials. We generalize or adapt some of the identities given by Bergum and Hoggatt [4,13] to our definition of the GFP. Most of the numerical identities for Fibonacci and Lucas numbers that we extend to GFP can be found in the books, articles or webpages on Fibonacci and Lucas numbers and their applications, [7,9,10,11,8,12].
We note that once we have an identity -from literature-for numerical sequences it is not too complicated to extend this to GFPs. However, the numerical identities do not extend automatically to GFPs; they need some adjustments (some of the identities in the paper were balanced using Mathematica R ). Therefore, the aim of this paper is to give a collection of identities for the GFP. In particular, the identities that we give here apply to the following familiar polynomial sequences: 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.

Generalized Fibonacci polynomials
In this section we introduce the generalized Fibonacci polynomial sequences. This definition gives rise to some familiar polynomial sequences which are mentioned below. The definition matches the definitions of polynomial sequences given in other papers on this topic, for example the definition given by Flórez et al. and Koshy [5,9], respectively.
For the remaining part of this section we reproduce the definitions by Flórez et al. [5] and [6], for generalized Fibonacci polynomials.
The generalized Fibonacci polynomial sequence, is defined by the recurrence relation where p 0 (x) is a constant and p 1 (x), d(x), and g(x) are non-zero polynomials in Z[x] with gcd(d(x), g(x)) = 1. If we classify this family of polynomials by the nature of their initial conditions, we obtain two distinguishable subfamilies. We say that a sequence is of Lucas type if 2p 1 (x) = p 0 (x)d(x) with |p 0 (x)| = 1 or 2 and we say that a sequence is of Fibonacci type if p 0 (x) = 0 with p 1 (x) = 1. Table 1 shows some familiar examples of those types of polynomial sequences.

Polynomial
Initial value Initial value Recursive Formula We denote the GFP of Lucas type and Fibonacci type by G * n (x) and G ′ n (x) respectively. We now give the Binet formulas for both types of GFPs namely, the Lucas and Fibonacci types that were proved by Flórez et al. [5]. The Binet formula for GFP of Lucas type is given by The Binet formula for GFP of Fibonacci type is given by are the polynomials defined on the generalized Fibonacci polynomial.
We say that a generalized Fibonacci sequence of Lucas (Fibonacci) type is equivalent to a sequence of the Fibonacci (Lucas) type, if their recursive sequences are determined by the same polynomials d(x) and g(x). Notice that two equivalent polynomials have the same a(x) and b(x) in their Binet representations (see Table 2).
Note. The definition of the Generalized Fibonacci Polynomial by Flórez et al. [6], differs from the definition in this paper due to the initial conditions of the Fibonacci type polynomials. Thus, the initial conditions for the Fibonacci type polynomials given by Flórez, Higuita, and Mukherjee, [6] are G 0 (x) = p 0 (x) = 1 and so implicitly G −1 (x) = 0. However, our definition for the Lucas type polynomial is identical to that found in the same article.

Identities
In this section we give a collection of identities for the GFP. These identities apply, in particular, to our familiar set of GFP. Thus, the identities apply 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. The identities below and their proofs are expressed in terms of α, d(x), g(x), a(x), and b(x). Table  2 gives those values for the familiar polynomials mentioned above. Note: For the sake of simplicity throughout the rest of this paper, we will use d, g, a, and b instead of d(x), g(x), a(x), and b(x) respectively.
For example, suppose we want to apply the identity in Proposition 3 part (1) , to Chebyshev polynomials of the first and second kind, namely T n (x) and U n (x) respectively. We begin by getting the appropriate information for U n (x) and T n (x) given in Table 2 x 2 − 1, α = 2, and g(x) = −1. Next, we observe that U n (x) is Fibonacci type, so G ′ n (x) equals U n (x) and from Table 2 line 4, we get that its equivalent Lucas type polynomial G * n (x) equals T n (x). Note that (a − b) 2 = 4x 2 − 4. Now substituting all this information into the identity given in Proposition 3 part (1) we obtain, (4x 2 − 4)U n (x) = 2 (T n+1 (x) + (−1)T n−1 (x)) , for all n.
If we want to apply the same identity for Jacobsthal J n (x), then we substitute the information given in Table 2  The expression (a − b) 2 appears very often in the following list of identities. It is easy to see that However, in the following list of identities we keep the factor (a − b) 2 as is. Proposition 2 parts (1) and (2) are generalizations of Cassini and Catalan identities respectively. One can refer to the book by Koshy [9] to learn more about them. We have found that all of the identities in this paper can be proved by substituting the Binet formula on each side of the proposed identity. Since the reader can check it without major difficulty, we prove some identities and leave the others as exercises.  (1). We prove this part using Binet formula (3) from Section 2. Therefore, substituting formula (3) Simplifying this expression, we obtain [2(ab) n − (a n+1 b n−1 + b n+1 a n−1 )] /(a − b) 2 . Factoring and simplifying further we see that the last expression reduces to −(ab) n−1 , and this is equal to (−1) n g n−1 .

Proof of part (2). Using Binet formula (3) in (G
Simplifying the right hand side of the last expression we obtain, Factoring the last expression, we get [(a n+m − b n+m )(a n−m − b n−m )] /(a − b) 2 , and this is equal to G ′ n+m (x)G ′ n−m (x).
(88) If m and r are positive integers and m > r, then (This is an adaptation of the Pythagorean Theorem for the Fibonacci type polynomials) (90) If n is a positive integers, then (dG * n+1 (x)) 2 + (G * n+2 (x)) 2 2 equals (This is an adaptation of the Pythagorean Theorem for the Lucas type polynomials) (91) If n and m are positive integers, then (92) If m and n are positive integers, then (94) If m and n are positive integers, then Proof of part (1). We know G ′ n (x) is a Fibonacci type polynomial, so we can represent it using Binet formula (3). Therefore, (a−b) 2 G ′ n (x) can be written as (a−b) 2 [(a n −b n )/(a−b)]. Simplifying, we see the above expression equals (a n+1 + b n+1 − ab n − ba n ). This reduces to a n+1 + b n+1 + g(a n−1 + b n−1 ), since we know that ab = −g. It is easy to see that a n+1 + b n+1 + g(a n−1 + b n−1 ) = α a n+1 + b n+1 α + g a n−1 + b n−1 α .
This and Binet formula (2) for Lucas type polynomials complete the proof.

Acknowledgement
The first and the last authors were partially supported by The Citadel Foundation.