Matrix Structure of Jacobsthal Numbers

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


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][2][3][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][6][7][8][9][10][11][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][14][15] and many others. The authors in [16,17] have defined the Jacobsthal numbers J n by the following recurrence relation: 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 v j , he proved that for It is obvious that the Jacobsthal sequence is a particular demonstration of a sequence given recursively as follows: where c 0 , c 1 , ⋯, c j−1 are real constants. The author in [10] has determined a closed-form formula for the generalized sequence by companion matrix method as follows: Then, by an inductive argument, the generalization of (3) will be obtained, viz., where a n 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: and proved for any natural numbers that Quite recently, for n > 0 and 1 ≤ r ≤ j, the authors in [19] have defined the j sequences of the generalized order j-Jacobsthal numbers as follows: with for 1 − j ≤ n ≤ 0 and J r n 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 for the generalized order j-Jacobsthal sequences, where C and is known as the generalized order j-Jacobsthal matrix.

Main Results
In this section, we present some new generalization of Jacobsthal numbers in the form of a matrix and prove some basic properties.
For different values of s, we have following deductions: Deduction 1. Choosing s = 0, the sequence w r n gets reduced to the generalized order-j Fibonacci sequence [25].

Journal of Function Spaces
We can redefine it by the vector recurrence relation as follows: for the generalized order j-Jacobsthal sequences, where C and this matrix C is known as the generalized order j-Jacobsthal matrix. We now define a j × j matrix B n = ðμ rm Þ as follows: Then, clearly, we get the following matrix equation by expanding (15) to j columns: In this direction, we have following result: Theorem 3. Define j × j matrices C and B n , respectively, given by (16) and (17); then for every n ≥ 0, we have Proof. Using (17), we see that B n = C · B n−1 . Now employing the inductive argument, we can write B n = C n−1 · B 1 . But by definition of generalized order j-Jacobsthal numbers, B 1 = C, and consequently, we have B n = C n , as required. ☐ Theorem 4. Define j × j matrx B n given by (17); then Proof. We know by using result Theorem 3 that B n = C n . Hence, we see that Consequently, the result follows by using Laplace expansion of determinants along any column. ☐ Lemma 5. Let w r n be the generalized order j-Jacobsthal number; then, Proof. We know by using Theorem 3 that B n = C n . Hence, we can write Consequently, the result follows by using Laplace expansion of determinants along any column. ☐

Generalized Binet Formula (GBF)
This part of the article deals with the derivation of GBF for generalized order j-Jacobsthal numbers. In 1843, it was Binet who derived the interesting formula using Fibonacci numbers: where the values of a and b are ð1 ± ffiffi ffi 5 p Þ/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 matrix C defined by (16) is 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. For s ≥ 0, the equation does not have multiple roots with j ≥ 3.
Let fðηÞ be the characteristic polynomial of the generalized order j-Jacobsthal matrix C. Let η 1 , η 2 , ⋯, η j be the eigenvalues of C, which are clearly all distinct by Lemma 6. Let j 3 Journal of Function Spaces × j Vandermonde matrix V be given by Also, let θ r j be a j × 1 matrix as follows: and V ðrÞ m be a j × j matrix obtained from V by replacing mth column of V by θ r j : Theorem 7. For 1 ≤ r ≤ j, let w r n be the nth term of rth Jacobsthal sequence; then, Proof. As we know that matrix C has all distinct eigenvalues, consequently, it is diagonalizable. Denote V T = Ω, and obviously, Ω is invertible; then, Ω −1 CΩ = E, where E is given by Consequently, C is similar to Ω and hence yields C n Ω = ΩC n . But using Theorem 3, we know that B n = C n . We thus have the following system of equations: where B = ðμ rm Þ j×j . Hence, for each m = 1, 2, ⋯, j giving by observing that μ rm = w m n−r+1 and the result follows. ☐

Data Availability
In this manuscript, we have approached generalized Binet formula in a different way, and we concerned j-Jacobsthal matrix given in the paper in a generalized way.

Conflicts of Interest
The authors declare that they have no conflicts of interest.