# Matrix Structure of Jacobsthal Numbers

Authors: Abdul Hamid Ganie; Mashael M. AlBaidani
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The main scenario of this paper is to introduce a new sequence of Jacobsthal type having a generalized orderj. Some basic properties will be studied concerning it. Also, we will establish the generalized Binet formula.


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## Body

## 1. Background and Introduction

The Fibonacci sequence is an integer sequence plays a vital role for many fascinating identities. In nature, it shows its presence, even if certain fruits are looked at, the number of little bumps around each ring is counted or the sand on the beach, and how waves hit it is watched out, the Fibonacci sequence is seen there. It was studied by many authors in the well-known systematic manner, and attractive investigations have been witnessed as can be seen in [1–4]. Further, several recurrence sequences of natural numbers have been object of study for many researchers. Illustrations of these are the Fibonacci, Lucas, Pell, Pell-Lucas, Modified Pell, Jacobsthal, and Jacobsthal-Lucas sequences among others as can be seen in [5–12].It is well known that the Jacobsthal numbers obey attracting structure in many fields of science, engineering and technology as can be seen in [13–15] and many others. The authors in [16, 17] have defined the Jacobsthal numbers Jn by the following recurrence relation:
(1)J0=0,J1=1,Jn+2=Jn+1+2Jn,n≥0.The author in [18] has shown that some interesting properties of Fibonacci sequence can be obtained from a matrix description. For a jth Fibonacci number vj, he proved that for
(2)A=0111that
(3)An01=vnvn+1.It is obvious that the Jacobsthal sequence is a particular demonstration of a sequence given recursively as follows:(4)ar+j=c0ar+c1ar+1+⋯+cj−1ar+j−1,where c0,c1,⋯,cj−1 are real constants. The author in [10] has determined a closed-form formula for the generalized sequence by companion matrix method as follows:
(5)010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01c0c1c2⋯cjcj−1.Then, by an inductive argument, the generalization of (3) will be obtained, viz.,
(6)Ana0a1⋮aj−1=anan+1⋮an+j−1,where an is the nth term of the sequence.It is well established fact that the linear recurrence relations play a vital role of number theory. They show their appearance in almost everywhere in mathematics and computer science as can be found in [17].In [15], the authors have studied Jacobsthal F-matrix as follows:
(7)F=1210,and proved for any natural numbers that
(8)Fn=Jn+12JnJn2Jn−1.Quite recently, forn>0 and 1≤r≤j, the authors in [19] have defined the j sequences of the generalized order j-Jacobsthal numbers as follows:
(9)Jnr=Jn−1r+2Jn−2r+Jn−3r+⋯+Jn−jr,with
(10)Jnr=1,ifr=1−n,0,otherwise,for 1−j≤n≤0 and Jnr is the nth term of the rth sequence and was shown that the fundamental recurrence relation (9) can be defined by the vector recurrence relation
(11)Jn+1rJnrJn−1r⋮Jn−j+2r=CJnrJn−1rJn−2r⋮Jn−j+1r,for the generalized order j-Jacobsthal sequences, where C(12)C=cijj×j=121⋯11100⋯00010⋯00001⋯00⋮⋮⋮⋱⋮⋮000⋯10and is known as the generalized orderj-Jacobsthal matrix.


## 2. Main Results

In this section, we present some new generalization of Jacobsthal numbers in the form of a matrix and prove some basic properties.Following the authors in [2, 5, 20–24], we define, for n>0, 1≤r≤j, and s≥0, the j sequences of the generalized order j-Jacobsthal numbers as follows:
(13)wnr=wn−1r+2swn−2r+wn−3r+⋯+wn−jr,with initial conditions as
(14)wnr=1,ifr=1−n,0,otherwise,for 1−j≤n≤0 and wnr is the nth term of the rth generalized Jacobsthal sequence.For different values ofs, we have following deductions:Deduction 1.
Choosings=0, the sequence wnr gets reduced to the generalized order-j Fibonacci sequence [25].Deduction 2.
Choosings=1, we get the results obtained in [19].
We can redefine it by the vector recurrence relation as follows:(15)wn+1rwnrwn−1r⋮wn−j+2r=Cwnrwn−1rwn−2r⋮wn−j+1r,for the generalized order j-Jacobsthal sequences, where C(16)C=cijj×j=12s1⋯11100⋯00010⋯00001⋯00⋮⋮⋮⋱⋮⋮000⋯10
and this matrixC is known as the generalized order j-Jacobsthal matrix. We now define a j×j matrix Bn=μrm as follows:
(17)Bn=wn1wn2⋯wnjwn−11wn−12⋯wn−1j⋮⋮⋮⋱wn−j+11wn−j+12⋯wn−j+1j.
Then, clearly, we get the following matrix equation by expanding (15) to j columns:
(18)Bn+1=C·Bn.In this direction, we have following result:Theorem 3.
Definej×j matrices C and Bn, respectively, given by (16) and (17); then for every n≥0, we have
(19)Bn=Cn.Proof.
Using (17), we see that Bn=C·Bn−1.
Now employing the inductive argument, we can writeBn=Cn−1·B1. But by definition of generalized order j-Jacobsthal numbers, B1=C, and consequently, we have Bn=Cn, as required.☐Theorem 4.
Definej×j matrx Bn given by (17); then
(20)detBn=−2s,ifj=2,1,ifjisodd,−1,ifjisevenwithk≠2.Proof.
We know by using result Theorem3 that Bn=Cn. Hence, we see that
(21)detBn=detCn=detCn.
Consequently, the result follows by using Laplace expansion of determinants along any column.☐Lemma 5.
Letwnr be the generalized order j-Jacobsthal number; then,
(22)wn+11=wn1+wn2,wn+12=2swn1+wn3,wn+1r=wn1+wnr+1;3≤r≤j−1,wn+11=wnj.Proof.
We know by using Theorem3 that Bn=Cn. Hence, we can write
(23)Bn=Bn−1B1.
Consequently, the result follows by using Laplace expansion of determinants along any column.☐


## 3. Generalized Binet Formula (GBF)

This part of the article deals with the derivation of GBF for generalized orderj-Jacobsthal numbers.In 1843, it was Binet who derived the interesting formula using Fibonacci numbers:(24)Fn=an−bna−b,where the values of a and b are 1±5/2. Moreover, the attractive Binet formula for generalized Fibonacci numbers is studied in [12].It is well know from the concept of companion matrices that the characteristic equation of the matrixC defined by (16) is
(25)ςj−ςj−1−2sςj−2−ςj−3−⋯−ς−1=0,which is also the characteristic equation of generalized order j-Jacobsthal numbers.In order to prove our main result of this section, we first define the following lemma without proof:Lemma 6.
Fors≥0, the equation
(26)ςj−ςj−1−2sςj−2−ςj−3−⋯−ς−1=0does not have multiple roots with j≥3.
Letfη be the characteristic polynomial of the generalized order j-Jacobsthal matrix C. Let η1,η2,⋯,ηj be the eigenvalues of C, which are clearly all distinct by Lemma 6. Let j×j Vandermonde matrix V be given by
(27)V=η1j−1η1j−2⋯η11η2j−1η2j−2⋯η21⋮⋮⋱⋮⋮ηjj−1ηjj−2⋯ηj1.
Also, letθjr be a j×1 matrix as follows:
(28)θjr=η1n+j−rη2n+j−r⋮ηjn+j−r.and Vmr be a j×j matrix obtained from V by replacing mth column of V by θjr.Theorem 7.
For1≤r≤j, let wnr be the nth term of rth Jacobsthal sequence; then,
(29)wn−r+1k=detVmrdetV.Proof.
As we know that matrixC has all distinct eigenvalues, consequently, it is diagonalizable. Denote VT=Ω, and obviously, Ω is invertible; then, Ω−1CΩ=E, where E is given by
(30)E=diagη1,η2,⋯,ηj.
Consequently,C is similar to Ω and hence yields CnΩ=ΩCn. But using Theorem 3, we know that Bn=Cn. We thus have the following system of equations:
(31)μr1η1j−1+μr2η1j−2+⋯+μrj=η1n+j−rμr1η2j−1+μr2η2j−2+⋯+μrj=η2n+j−r⋮μr1ηjj−1+μr2ηjj−2+⋯+μrj=ηjn+j−r,where B=μrmj×j. Hence, for each m=1,2,⋯,j giving
(32)μrm=detVmrdetV,by observing that μrm=wn−r+1m and the result follows.☐



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