On The Lagrange Interpolation of Fibonacci Sequence

Fibonacci sequence is one of the most common sequences in mathematics. It was first introduced by Leonardo Pisa in his book Liber Abaci (1202). From the first n+ 1 terms of Fibonacci sequence, a polynomial of degree at most n can be constructed using Lagrange interpolation. In this paper, we show that this Fibonacci Lagrange Interpolation Polynomial (FLIP) can be obtained both recursively and implicitly.


I. INTRODUCTION
A T first, Fibonacci sequence is the solution of the pairs of rabbits problem [1] "In the beginning we only have one pair of rabbits (male and female).How many pairs of rabbits we have after one year if every month each pair begets a new pair which from the second month on becomes productive?" After many years passed, Fibonacci sequence surprisingly can be found in nature such as in nautilus shells, pine cones and sunflowers [1].In mathematics, Fibonacci occurs in many fields for instance in linear algebra, combinatorics, and discrete mathematics.There are some developments about Fibonacci sequence, for example Fibonomial (Fibonacci and Binomial) [2], [3] and Fibonacci Polynomial [4], [3].
The study of polynomials generated by Fibonacci sequence is quite interesting.Fibonacci Polynomials (FP) are generated recursively similar to Fibonacci sequence [3] while Fibonacci-Coefficient Polynomials (FCP) are generated by putting Fibonacci sequence as coefficient [4].In this paper, we propose a novel approach to construct polynomial based on Fibonacci sequence.The idea is by locating Fibonacci sequences as points in coordinate system and then find a polynomial that passes through those points using Lagrange Interpolation.
The outline of this paper is as follows.In Section 2 the preliminary of this paper is given.Section 3 defines the Lagrange interpolation of Fibonacci sequence to get FLIP.Hereafter, we show that FLIP can be obtained both recursively and implicitly.Finally, the conclusion of this paper is given in Section 4.

II. NOTATIONS AND PRELIMINARIES
We denote N as the set natural number and n = {0, 1, 2, . . ., n}.We also use combinatorial notations such as and a binomial identity A. Fibonacci Sequence Definition 1 (Fibonacci Sequence [3]): Fibonacci sequence is a recursive sequence defined by with The implicit formula of Fibonacci sequence is where ϕ = 1+ √ 5 2 and ϕ + υ = 1.The irrational number ϕ is well known as golden section or golden ratio.Both ϕ and υ are the roots of x 2 − x − 1 = 0.

B. Lagrange Interpolation
In the field of numerical analysis, interpolation is a process finding the most fitted function from some specified points.The simplest approximation of an interpolation is a polynomial [5].It means that given some points, there exists a polynomial that passes through that points.And the polynomial is close to the function.One of the polynomial interpolation methods is Lagrange Interpolation.Suppose we have n + 1 different points (x i , y i ), i ∈ n.We need find a polynomial L(x) that satisfies L(x i ) = y i for all i ∈ n.The basic idea of Lagrange Interpolation is constructing the basis polynomials l n,i (x) such that l n,i ( for all i, j ∈ n.By considering all factors (x − x i ) for i ∈ n, we can construct l n,i (x) as follows After constructing these basis polynomials, we can generate the Lagrange Interpolation polynomial as follows

III. INTERPOLATING THE FIBONACCI SEQUENCE
Before starting the interpolation, we put the sequences in the xy-coordinate system.Let denote Fibonacci point p n = (n, f n ) as the point from the n th term of Fibonacci sequence.As example, the Fibonacci points for n = 0 until n = 4 are shown in Fig. 1.In this paper, we define FLIP n (x) as the polynomial that generated using Lagrange Interpolation from p i for i ∈ n.In this case, we will interpolate from points (x i , y i ) = (i, f i ).Consequently, we have x i = i and y i = f i in ( 5) and then we can write Surprisingly, (6) can be simplified because of (i − j) factors.Let define (n, i) as follows By the definition, (n, i) is the product of n non-zero consecutive integers in which there are i positive integers.Therefore, we get By substituting (7) to (6), we have As example, from (8) we can derive FLIP n (x) for n = 1, 2, 3, 4 as follows The graphics of above polynomials are shown in Fig. 2. We should remember that FLIP n (i) = f i for i ∈ n.Before deriving the other formulas of FLIP n (x), we will prove a theorem about its leading coefficient.
Theorem 1: The leading coefficient of FLIP n (x) is . Proof: The leading coefficient of FLIP n (x) is equal to

A. Recursive and Implicit Formulas
In this part, we will derive the other formulas of FLIP n (x).It is quite impressive that FLIP n (x) can be generated recursively and implicitly, like Fibonacci sequence.