Research Article

Matrix Representation of an All-inclusive Fibonacci Sequence

Abstract

Background and Objectives: Fibonacci sequence is a sequence of positive integers that has been studied over several years. The aim of this paper was to suggest new generalized Fibonacci sequence to a particular class of recursive sequence. Materials and Methods: The equilibrium point of the model was investigated and a new sequence. The matrix method was applied to perform the generalization. Results: The nth power of the generating matrix for this generalized Fibonacci sequence was established and some basic properties of this sequence were obtained by matrix methods. Conclusion: Cassini’s identity and Binet’s formula for the generalized Fibonacci sequence was obtained.

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Tanveer Ahmad Tarray, Parvaiz Ahmad Naik and Riyaz Ahmad Najar, 2018. Matrix Representation of an All-inclusive Fibonacci Sequence. Asian Journal of Mathematics & Statistics, 11: 18-26.

DOI: 10.3923/ajms.2018.18.26

INTRODUCTION

The Fibonacci sequence is a series of numbers in which a number is found by adding the two numbers before it. Initially with 0 and 1, the sequence starts from 0, 1, 1, 2, 3, 5, 8... by the rule the expression is Xn = Xn-1+Xn-2 which was named after Fibonacci also known as Leonardo of Pisa or Leonardo Pisano and first introduced by Liber abaci in 1202. Knowledge of numbers is said to have first originated in the Hindu-Arabic arithmetic system, which Fibonacci studied while growing up in North Africa. The well-known Fibonacci sequence is a sequence of positive integers that has been studied over several years. The most and vast research field of Fibonacci numbers is defined to study the generalizations of Fibonacci numbers Bilgici1 and Tasyurdu et al.2. The main aim of the present paper is to study other generalized Fibonacci sequence by matrix methods.

Horadam3 introduced and studied the generalized Fibonacci sequence Wn = Wn(a,b;p,q) defined by: where, a, b, p and q are arbitrary complex numbers with q≠0. These numbers were first studied by Horadam3 and are called Horadam numbers. In Silvester4, it has been shown that a number of the properties of the Fibonacci sequence can be derived from a matrix representation. In doing so, it showed that if: then: where, un represents the nth Fibonacci number. In Koken and Bozkurt obtained some important properties of Jacobsthal numbers by matrix methods, using diagonalization of a 2×2 matrix to obtain a Binet’s formula for the Jacobsthal numbers and in that study, 2×2 matrix and its nth power are defined respectively as: and: where, Jn is the nth Jacobsthal number. In Demirturk5 obtained summation formulae for the Fibonacci and Lucas sequences by matrix methods. For doing this, it considered 2×2 matrix such as: and: where, Fn and Ln are nth Fibonacci and Lucas numbers respectively. The authors presented some important relationship between k-Jacobsthal matrix sequence and k-Jacobsthal-Lucas matrix sequence and k is the positive real number6. In Godase and Dhakne7 described some properties of k-Fibonacci and k- Lucas number by matrix terminology. To obtain such properties , the authors 2×2 matrix such as: and: where, k is the fixed positive real number. Catarino and Vaso8 obtained 2×2 matrix for the k-Pell sequence with in nth power and Catarino9 presented Binet’s formula for the k-Pell sequence by the diagonalization of 2×2 matrix. In both studies of Catarino9 and Ugyun and Eldogm6 defined 2×2 matrix as such as: and: where, Pk,n is the nth k-Pell number. Again in Catarino and Vaso8 study, 2×2 matrix they have obtained Binet’s formulae for the k-Pell-Lucas sequence by the matrix diagonalization and also obtained some properties of k-Pell Lucas sequence with the help of a 2×2 matrix as such as: and: where, Qk,n is the nth k-Pell Lucas number. Borges et al.10 have used the same concept as in Catarino9 and studied k-Pell-Lucas sequence by matrix methods.

Preliminaries: In the study of Catarino9 for the positive real number the k-Pell sequence {Pk,n} is defined by the recurrence relation: (1)

Again in the study of Catarino9, the positive real number k the k-Pell-Lucas sequence {Qk,n} is defined recurrently by: (2)

The sequence of Eq. 1 and 2 have the same characteristics equation x2-2x-k = 0 and let a and b are the roots of the characteristic equation. The well-known general forms of both sequences known as Binet’s Formulae are given and written by: and Qk,n = an+bn where, .

The main aim of this paper was to study other generalized Fibonacci sequences by matrix methods.

MATERIALS AND METHODS

Definition: For the positive real number k, the generalized Fibonacci sequence, say Sk,n defined by: (3)

Clearly x2-2x-k = 0 is also the characteristic equation of the sequence (Eq. 3). It produces two roots as: (4)

Also the 2×2 matrix called generating matrix for the sequence (Eq. 3) is defined as: (5)

RESULTS

Theorem 1: Binet Formulae for the generalized Fibonacci sequence: (6)

and: (7)

Proof: The general form for the generalized Fibonacci sequence may be expressed in the form: where, A and B are constants that can be determined by the initial conditions. So put the values n = 0 and n = 1 in Eq. 3, we get: A+B = 1 and Aa+Bb = 1.

After solving the above system of equations for A and b, we get: Therefore: And by Eq. 4, we have: This proves the first part of the theorem.

Now if we consider Eq. 4 and above proof, we get: This proves the second part of the theorem.

Theorem 2: for n∈N, we have: (8) (9)

Proof: To prove this we will use Eq. 6, 7, 1 and 3:  which proves Eq. 8.

Now: which proves Eq. 9.

Theorem 3: the nth power of the generating matrix. for n∈N, we have: (10)

Proof: Here we shall use induction on n. Indeed Eq. 10 is true for n = 1. Now suppose that Eq. 10 is true for n. let us show that the result is true for n+1, then:  as required.

Theorem 4: (Cassini’s Identity) for n∈N, we have: (11)

Proof: From Eq. 5, det. (S)n = (-k)n and now from Eq. 10, we have: then: (12)

Put: and then:  Therefore: Since from Eq. 5, det(Sn) = (-k)n, then: Hence the result.

From Eq. 12 in the proof of the above theorem we also conclude that: (13)

Theorem 5: For n∈N we get: Proof: To prove the ongoing result we shall use introduction on n. indeed the result is true for n = 1. Suppose that the result is true for n, then: (14)

as desired.

Theorem 6: For n>0, we get: (15)

Proof: To prove the ongoing result we shall use induction on n. indeed the result is true for n = 0, suppose that the result is true for n then: Since the result is true for n then: as desired.

Binet’s formula by matrix diagonalization of generating matrix: Use the diagonalization of the generating matrix (Eq. 5) to obtain Binet’s formula for the generalized Fibonacci sequence (Eq. 3). For this purpose we should prove the following theorem:

Theorem 6: For n>0, we get: (16)

Proof: The generating matrix is given by: now here our motive is to diagonalize the generating matrix S. Since S a square matrix and so let x be the eigen value of S and then by the Cayley Hamilton theorem on matrix, we get: This is the characteristic equation of the generating matrix. Let a and b be the roots of the characteristic equation and also a and b are two eigen values of the square matrix S. Now we will try to find the eigen vectors corresponding to the eigen values a and b. to find the eigen vectors we simply solve the system of linear equations given by:

 |S-xI|V = 0

where, V is the column vector of order 2×1. First of all we calculated the eigen vectors corresponding to the eigen value of a and then: Consider the system: And if we take V2 = t in (17) , we get, V1 = at. Hence the eigen vectors corresponding to a are of type: In particular t = 1, the eigen vector corresponding to a are of type: Similarly the eigen vector corresponding to b is: Let A be the matrix of eigenvectors: so and then: Now we keep the diagonal matrix D in which eigen values of S are on the main diagonal: and then by the diagonalization of matrix, we get: By using Eq. 15, we have: Let and using a+b = 2, we achieve: Therefore, by equating corresponding terms on both sides we get: which proves Eq. 16.

Theorem 7: The generalized characteristic roots of Sn are: (17)

and: (18)

Proof: if we write the characteristic polynomial of Sn, we achieve: After using equations Eq. 9, 11 and 13, we conclude that: Hence the characteristic equation of Sn is given by: And the generalized characteristic roots are obtained from: Clearly the above equation has roots given an and bn and consequently we get the desired results as: And: Hence the result.

Theorem 8: The characteristic equation of S is: (19)

Proof: Here we employ the method of matrices as well as determinants to obtain the characteristic equation for S. Eq. 10 gives: Since the ratio of two consecutive generalized Fibonacci numbers is equal to a, then: and: Again: Therefore: If we consider Eq. 4, we have: If we will compute the determinants of both sides, we get, the characteristic equation of the matrix S as follow: as required.

DISCUSSION

The main aim of the present paper was to study generalized Fibonacci sequence by matrix methods. It has been shown theoretically that the proposed model is more efficient than the existing models applying matrix method. The results obtained in this study are in close agreement with the previous existing studies11-17.

CONCLUSION

In this study the nth power of generalized Fibonacci sequence was established and some fundamental properties of this sequence were attained by matrix methods and Cassini’s identity and Binet’s formula for the generalized Fibonacci sequence was achieved.

SIGNIFICANCE STATEMENTS

It has been proposed some important relationship between k-Jacobsthal matrix sequence and k-Jacobsthal-Lucas matrix sequence and k is the positive real number. Godase and Dhakne described some properties of k-Fibonacci and k- Lucas number by matrix terminology. This study will help the researchers to uncover the critical areas related to Fibonacci sequence to a particular class of recursive sequence. For the future research, researcher can be considering a new theory for Fibonacci sequence.

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