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Matrix Representation of an All-inclusive Fibonacci Sequence



Tanveer Ahmad Tarray, Parvaiz Ahmad Naik and Riyaz Ahmad Najar
 
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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

URL: https://scialert.net/abstract/?doi=ajms.2018.18.26
 
Received: September 14, 2018; Accepted: September 24, 2018; Published: November 05, 2018


Copyright: © 2018. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

then:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(1)

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

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and Qk,n = an+bn where, Image for - Matrix Representation of an All-inclusive Fibonacci Sequence.

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(3)

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

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(4)

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

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(5)

RESULTS

Theorem 1: Binet Formulae for the generalized Fibonacci sequence:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(6)

and:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(7)

Proof: The general form for the generalized Fibonacci sequence may be expressed in the form:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Therefore:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

And by Eq. 4, we have:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

This proves the first part of the theorem.

Now if we consider Eq. 4 and above proof, we get:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

This proves the second part of the theorem.

Theorem 2: for n∈N, we have:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(8)

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(9)

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

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

which proves Eq. 8.

Now:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

which proves Eq. 9.

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

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

as required.

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

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(11)

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

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

then:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(12)

Put:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and then:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Therefore:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Since from Eq. 5, det(Sn) = (-k)n, then:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Hence the result.

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

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(13)

Theorem 5: For n∈N we get:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(14)

as desired.

Theorem 6: For n>0, we get:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Since the result is true for n then:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(16)

Proof: The generating matrix is given by:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Consider the system:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

And if we take V2 = t in (17) , we get, V1 = at. Hence the eigen vectors corresponding to a are of type:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

In particular t = 1, the eigen vector corresponding to a are of type:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Similarly the eigen vector corresponding to b is:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Let A be the matrix of eigenvectors:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

so and then:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Now we keep the diagonal matrix D in which eigen values of S are on the main diagonal:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and then by the diagonalization of matrix, we get:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

By using Eq. 15, we have:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Let Image for - Matrix Representation of an All-inclusive Fibonacci Sequence and using a+b = 2, we achieve:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Therefore, by equating corresponding terms on both sides we get:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

which proves Eq. 16.

Theorem 7: The generalized characteristic roots of Sn are:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(17)

and:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(18)

Proof: if we write the characteristic polynomial of Sn, we achieve:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

After using equations Eq. 9, 11 and 13, we conclude that:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Hence the characteristic equation of Sn is given by:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

And the generalized characteristic roots are obtained from:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Clearly the above equation has roots given an and bn and consequently we get the desired results as:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

And:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Hence the result.

Theorem 8: The characteristic equation of S is:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence
(19)

Proof: Here we employ the method of matrices as well as determinants to obtain the characteristic equation for S. Eq. 10 gives:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Since the ratio of two consecutive generalized Fibonacci numbers is equal to a, then:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

and:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Again:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

Therefore:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

If we consider Eq. 4, we have:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

If we will compute the determinants of both sides, we get, the characteristic equation of the matrix S as follow:

Image for - Matrix Representation of an All-inclusive Fibonacci Sequence

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.

REFERENCES

1:  Bilgici, G., 2014. New generalizations of Fibonacci and Lucas sequences. Applied Math. Sci., 8: 1429-1437.
CrossRef  |  Direct Link  |  

2:  Tasyurdu, Y., N. Cobanoglu and Z. Dilmen, 2016. On the a new family of k-Fibonacci numbers. J. Sci. Technol., 9: 95-101.
Direct Link  |  

3:  Horadam, A.F., 1965. Basic properties of a certain generalized sequence of numbers. Fibonacci Q., 3: 161-176.
Direct Link  |  

4:  Silvester, J.R., 1979. Fibonacci properties by matrix methods. Math. Gaze., 63: 188-191.
CrossRef  |  Direct Link  |  

5:  Demirturk, B., 2010. Fibonacci and Lucas sums by matrix methods. Int. Math. Forum, 5: 99-107.
Direct Link  |  

6:  Uygun, S. and H. Eldogan, 2016. k-Jacobsthal and k-Jacobsthal Lucas matrix sequences. Int. Math. Forum, 11: 145-154.
CrossRef  |  Direct Link  |  

7:  Godase, A.D. and M.B. Dhakne, 2014. On the properties of k-Fibonacci and k-Lucas numbers. Int. J. Adv. Applied Math. Mech., 2: 100-106.
Direct Link  |  

8:  Catarino, P. and P. Vasco, 2013. Some basic properties and a two-by-two matrix involving the k-Pell numbers. Int. J. Math. Anal., 7: 2209-2215.
CrossRef  |  Direct Link  |  

9:  Catarino, P., 2013. A note involving two-by-two matrices of the k-Pell and k-Pell-Lucas sequences. Int. Math. Forum, 8: 1561-1568.
CrossRef  |  Direct Link  |  

10:  Borges, A., P. Catarino, A.P. Aires, P. Vasco and H. Campos, 2014. Two-by-two matrices involving k-Fibonacci and k-Lucas sequences. Applied Math. Sci., 8: 1659-1666.
CrossRef  |  Direct Link  |  

11:  Wloch, I., U. Bednarz, D. Brod, A. Wloch and M. Wolowiec-Musial, 2013. On a new type of distance Fibonacci numbers. Discrete Applied Math., 161: 2695-2701.
CrossRef  |  Direct Link  |  

12:  Wani, A.A., V.H. Badshah, S. Halici and P. Catarino, 2018. On a fibonacci-like sequence associated with k-lucas sequence. Acta Univ. Apulensis, 53: 41-54.

13:  Wani, A.A., G.P.S. Rathore, V.H. Badshah and K. Sisodiya, 2018. A two-by-two matrix representation of a generalized fibonacci sequence. Hacettepe J. Math. Statist., 47: 637-648.
Direct Link  |  

14:  Rathore, G.P.S., A.A. Wani and K. Sisodiya, 2016. Matrix representation of generalized k-fibonacci sequence. IOSR J. Math., 12: 67-72.
Direct Link  |  

15:  Wani, A.A., G.P.S. Rathore and K. Sisodiya, 2017. On generalizedk-fibonacci sequence by two-cross-two matrix. Global J. Math. Anal., 5: 1-5.
CrossRef  |  Direct Link  |  

16:  Naik, P.A. and K.R. Pardasani, 2018. Three-dimensional finite element model to study effect of RyR calcium channel, ER leak and SERCA pump on calcium distribution in oocyte cell. Int. J. Comput. Methods, Vol. 15, No. 6.
CrossRef  |  Direct Link  |  

17:  Naik, P.A. and K.R. Pardasani, 2014. Finite element model to study effect of Na+/K+ Pump and Na+/Ca2+ exchanger on calcium distribution in oocytes in presence of buffers. Asian J. Math. Stat., 7: 21-28.
CrossRef  |  Direct Link  |  

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