Palestine Journal of Mathematics

Vol. 11(1)(2022) , 292–304 © Palestine Polytechnic University-PPU 2022

MATRIX SEQUENCE OF THE BINOMIAL FORM OF THE

COMPLEX COMBINED JACOBSTHAL-AKIN SEQUENCE
A. A. Wani, G. C. Morales and N. A. Malik
Communicated by Ayman Badawi

MSC 2010 Classifications: Primary 11B37, 11B39; Secondary 11B83, 15A15, 05A10.

Keywords and phrases: Jacobsthal sequences, Jacobsthal-Like sequence, binomial form and matrix sequence.

Abstract In this note, we consider a complex sequence hUn in∈N which have a similar struc-
ture with Jacobsthal sequence and we nominate this sequence as a complex combined Jacobsthal-
Akin sequence or simply combined Jacobsthal-Akin sequence. After that we study a binomial
form hXn in∈N of hUn in∈N and we call hXn in∈N as binomial sequence. Finally we delineate a
matrix sequence hZn in∈N of the binomial sequence hXn in∈N .

1 Introduction
Many more authors worked on the generalizations of Fibonacci sequences [1] by various angles
or patterns. Especially some of them employed matrix methods as well as introduced complex
plane concept for the study of generalizations of Fibonacci numbers.
Horadam [2] in 1963 introduced the concept of complex Fibonacci numbers. Jordan [3] in
1965 considered a Gaussian Fibonacci sequence hGFn i and established some results between
Gaussian Fibonacci sequence and classical Fibonacci sequence. Gaussian Fibonacci numbers
are recursively defined by

GFn = GFn−1 + GFn−2 , n ≥ 2 and GF0 = i, GF1 = 1 (1.1)

Later on Berzsenyi [4], Harman [5] and Pethe [6] used different approaches of extensions of
Fibonacci numbers on the complex plane.
Now we give some literature where the authors studied the generalizations of Jacobsthal num-
bers and Jacobsthal-Lucas numbers (see [7]). Asci and Gurel [7] delineated and studied Gaussian
Jacobsthal and Gaussian Jacobsthal-Lucas numbers. These numbers are given, respectively, as
i
GJn+1 = GJn + 2GJn−1 , n ≥ 1 and GJ0 = , GJ1 = 1 (1.2)
2
i
Gjn+1 = Gjn + 2Gjn−1 , n ≥ 1 and Gj0 = 2 − , Gj1 = 1 + 2i. (1.3)
2
Again Asci ans Gurel [8] examined Gaussian Jacobsthal and Gaussian Jacobsthal-Lucas poly-
nomials.
In [10] defined a sequence hbn in∈Z0 (Z0 is the set of non-negative numbers) as the binomial
transform of the sequence han in∈Z0 if
n
X
bn = ak (1.4)
k=0

and Wani et al. [11] obtained the binomial form of Fibonacci-Like sequence.. A trend has been
going on from several past years that many authors added parameters s and t in the classical
Fibonacci, Jacobsthal sequences etc in the recurrence relation system of these sequences and
then designate these sequences as (s, t)-type sequences. Uygun [9] presented (s, t)-Jacobsthal
sequence h̂n (s, t)i and (s, t)-Jacobsthal-Lucas sequence hĉn (s, t)i such that

̂n (s, t) = s̂n−1 (s, t) + 2t̂n−2 (s, t) , n ≥ 2 and ̂0 (s, t) = 0, ̂1 (s, t) = 1 (1.5)
 MATRIX SEQUENCE OF THE BINOMIAL FORM 293

ĉn (s, t) = sĉn−1 (s, t) + 2tĉn−2 (s, t) , n ≥ 2 and ĉ0 (s, t) = 2, ĉ1 (s, t) = s (1.6)
where s > 0, t 6= 0 and s2 + 8t > 0.
Since 2008 several authors explored the recurrence relations of Fibonacci sequences, general-
ized Fibonacci sequences and other second order sequences into the sequences known as matrix
sequences that is, the sequences in which the terms of the sequences are in the form of matrices
and the elements of matrices are the terms of general sequences. In 2008 Civciv and and Turk-
men [12] presented (s, t)-Fibonacci sequence hFn (s, t)i and (s, t)-Fibonacci matrix sequence
hFn (s, t)i and obtained various properties for these sequences. These sequences hFn (s, t)i and
hFn (s, t)i are delineated by
Fn+1 (s, t) = sFn (s, t) + tFn−1 (s, t) n ≥ 1 and F0 (s, t) = 0, F1 (s, t) = 1 (1.7)

Fn+1 (s, t) = sFn (s, t) + tFn−1 (s, t) , n ≥ 1 (1.8)

" # " #
1 0 s 1
with F0 (s, t) = , F1 (s, t) = and s > 0, t 6= 0, s2 + 4t > 0.
0 1 t 0
Again Civciv and Turkmen [13] delineated (s, t)-Lucas matrix sequence which is defined as
follows:
Fn+1 (s, t) = sFn (s, t) + tFn−1 (s, t) n ≥ 1 and F0 (s, t) = 0, F1 (s, t) = 1 (1.9)

Ln+1 (s, t) = sLn (s, t) + tLn−1 (s, t) , n ≥ 1 (1.10)

" # " 2 #
s 2 s + 2t s
with L0 (s, t) = , L1 (s, t) = and s > 0, t 6= 0, s2 + 4t > 0.
2t −s st 2t
The main motive of this article to obtain the matrix sequence of the binomial form of the Com-
bined Jacobsthal-Akin sequence.

2 Combined Jacobsthal-Akin Sequence

Definition 2.1. [7] The Jacobsthal and Jacobsthal-Lucas sequences hJn i and hjn i are respec-
tively given by the following recurrence relations:
Jn = Jn−1 + 2Jn−2 , n ≥ 2 and J0 = 0, J1 = 1 (2.1)
jn = jn−1 + 2jn−2 , n ≥ 2 and J0 = 2, J1 = 1 (2.2)
The nth terms of both the sequences are mentioned by the ensuing relations:
αn − β n
Jn = (2.3)
α−β
jn = α n + β n (2.4)
where α = 2 and β = −1.
√
Definition 2.2. For s, t ∈ Z+ and i =


−1 , the combined Jacobsthal-Akin sequence hUn in∈N
is recurrently defined by
Un = iUn−1 + 2Un−2 , n ≥ 2 (2.5)
with seeds U0 = s − 2t and U1 = i (s − t)
The first few terms of the the combined Jacobsthal-Akin sequence hUn in∈N are given by
U0 = s − 2t, U1 = i (s − t) , U2 = s − 3t, U3 = i (3s − 5t) , U4 = − (s + t)
and so on.
Let θ and ϑ be the two complex roots of the characteristic equation u2 − iu − 2 = 0 of hUn i. The
values of θ and ϑ are determined by
√
√

7+i − 7−i
θ= and ϑ = (2.6)
2 2
294 A. A. Wani, G. C. Morales and N. A. Malik

Theorem 2.3. For n ∈ Z0 , the nth term of the combined Jacobsthal-Akin sequence is delineated
as
θn+1 − ϑn+1
Un = s − t (θn + ϑn ) (2.7)
θ−ϑ
Proof. Its proof can be easily seen by using induction method.
Now let
n+1
bn = θ − ϑn+1
U (2.8)
θ−ϑ
is called Jacobsthal-Like sequence.
and

U n = θ n + ϑn (2.9)

is called Jacobsthal-Lucas-Like sequence.

Clearly from the equations (2.8) and (2.9) Jacobsthal-Akin sequence hUn i is the combination of
two sequences such as jacobsthal-Like sequence (2.8) and Jacobsthal-Lucas-Like sequence (2.9)
and so the sequence hUn i is called combined Jacobsthal-Akin sequence.

3 Binomial Form of Combined Jacobsthal-Akin sequence hWn i

In the present section first of all we express combined Jacobsthal-Akin sequence hUn i in terms
of binomial form hXn i and we call hXn i as binomial sequence. After that we obtain a recur-
rence relation for hXn i. Furthermore we obtain binomial forms or binomial sequences of the
Jacobsthal-Like and Jacobsthal-Lucas-Like sequences.
Definition 3.1. For n ∈ Z0 , the binomial form of the combined Jacobsthal-Like sequence hUn i
is defined by
n  
X n
Xn = Ul (3.1)
l
l=0

Lemma 3.2. For n ∈ Z0 , the following property holds for hXn i:

n
!
X n

Xn+1 = Ul + Ul+1 (3.2)
l=0 l
! ! !
n+1 n n
Proof. Its proof can be easily obtained by using the relation = +
l l l−1
√
Theorem 3.3. (Recurrence relation for hUn i) For s, t ∈ Z+ and i = −1 , the binomial

recurrence relation hXn i of the combined Jacobsthal-Akin sequence hUn i is given by

Xn+1 = (2 + i) Xn + (1 − di) Xn−1 , n ≥ 1 (3.3)

with X0 = s − 2t and X1 = (s − 2t) + i (s − t)

Proof. Since
n
!
X n

Xn+1 = Ul + Ul+1
l=0 l
n
!
X n

= U0 + U1 + Ul + Ul+1
l=1 l
 MATRIX SEQUENCE OF THE BINOMIAL FORM 295

n
!
X n

= U0 + U1 + Ul + iUl + 2Ul−1 By Eqn. (2.5)
l=1 l
n
!
X n h i
= U0 + U1 + (1 + i) Ul + 2Ul−1
l=1 l
n
! n
!
X n X n
= (1 + i) Ul + 2 Ul−1 + U0 + U1
l=1 l l=1 l
n
! n
!
X n X n
= (1 + i) Ul + (1 + i) U0 + 2 Ul−1 − (1 + i) U0 + U0
l=1 l l=1 l
+ U1
n
! n
!
X n X n
= (1 + i) Ul + 2 Ul−1 − iU0 + U1
l=0 l l=1 l
n
!
X n
= (1 + i) Xn + 2 Ul−1 − iU0 + U1 By Eqn. (3.1) (3.4)
l=1 l

By replacing n by n − 1, we get
!
X1
n−
n−1
Xn = (1 + i) Xn−1 + 2 Ul−1 − iU0 + U1
l=1 l
! !
X1
n−
n−1 X1
n−
n−1
= iXn−1 + Ul + 2 Ul−1 − iU0 + U1
l=0 l l=1 l
n
! " ! ! !
X n−1 n−1 n−1 n−1
= iXn−1 + Ul−1 + 2 U0 + U1 +
l=1 l−1 1 2 3
! ! #
n−1 n−1
U2 + · · · + Un−2 + Un−1 − iU0 + U1
n−1 n
!
n−1
After using the fact = 0, we have
n

n
! n
!
X n−1 X n−1
Xn = iXn−1 + Ul−1 + 2 Ul−1 − iU0 + U1
l=1 l−1 l=1 l
n
" ! !#
X n−1 n−1
Xn = iXn−1 + +2 Ul−1 − iU0 + U1
l=1 l−1 l
n
" ! ! ! !#
X n−1 n−1 n−1 n−1
= iXn−1 + +2 +2 −2 Ul−1
l=1 l−1 l l−1 l−1
− iU0 + U1
n
" ! !#
X n−1 n
= iXn−1 + (1 − 2) +2 Ul−1 − iU0 + U1
l=1 l−1 l
n
! n
!
X n−1 X n
= iXn−1 − Ul−1 + 2 Ul−1 − iU0 + U1
l=1 l−1 l=1 l
296 A. A. Wani, G. C. Morales and N. A. Malik

! !
X1
n−
n−1 n
X n
= iXn−1 − Ul + 2 Ul−1 − iU0 + U1
l=0 l l=1 l
n
!
X n
= iXn−1 − Xn−1 + 2 Ul−1 − iU0 + U1 By Eqn. (3.1)
l=1 l
n
!
X n
= (i − 1) Xn−1 + 2 Ul−1 − iU0 + U1
l=1 l

Thus
n
!
X n
Xn − (i − 1) Xn−1 = 2 Ul−1 − iU0 + U1
l=1 l

Hence from the equation (3.4), we get

Xn+1 = (1 + i) Xn + Xn − (i − 1) Xn−1
= (2 + i) Xn + (1 − i) Xn−1
as required.
First few terms of the binomial sequence hXn i defined in equation (3.3) are as under
X0 = (s − 2t) , X1 = (s − 2t) + i (s − t) , X2 = (2s − 6t) + i (2s − 2t) , X3 = (4s − 13t) +
i (6s − 9t) , X4 = (6s − 25t) + i (16s − 27t) and so on.
Clearly v 2 − (2 + i) v − (1 − i) = 0 is the characteristic equation of hXn i . Suppose that γ and
δ be its two roots and are given as
q
2
(2 + i) + (2 + i) + 4 (1 − i)
γ=
2
√
7+i+2
=
2
√
7+i
= +1
2
=θ+1 (3.5)
Similarly
δ =ϑ+1 (3.6)
Some noticeable points about γ and δ are
√
γ + δ = 2 + i, γδ = i − 1 = − (1 − i) and γ − δ = 7 (3.7)

Now to obtain the binomial forms or binomial sequences of the Jacobsthal-Like hU

bn i and
Jacobsthal-Lucas-Like hU n i sequences we should prove the following result:
Theorem 3.4. For n ∈ Z0 , the nth term of hXn i is given by
!
γ n+1 − δ n+1 γ n − δn  
Xn = s − − t γ n + δn (3.8)
γ−δ γ−δ
" #
2+i 1−i
Proof. Let us consider a square matrix X = and u be the eigenvalue of X . Then
1 0
by Cayley Hamilton theorem the characteristic equation of X is given by the equation:

X − uI = 0
 MATRIX SEQUENCE OF THE BINOMIAL FORM 297

2+i−u 1−i
=0
1 u
u2 − (2 + i) u − (1 − i) = 0

Let γ and δ be the characteristic

" #
roots as"well
#
as eigenvalues of the matrix
"
X . The eigenvectors
#
γ δ γ δ
corresponding to γ and δ are and respectively. Let V1 = be the matrix of
1 1 1 1
" # " #
γ δ
−1 1 −δ
the eigenvectors. Since your matrix V1 = , the its inverse is γ − δ and
1 1 −1 γ
" #
γ 0
V2 = is the diagonal matrix. Then by the process of diagonalization of matrices, we
0 δ
achieve

X n = V1 V2n V1−1
" #" n #" #
−1
γ δ γ 0 1 −γ
= (γ − δ )
1 1 0 δn −1 δ
" n+1
− δ n+1 −δγ n+1 + γδ n+1
#
−1
γ
= (γ − δ )
γ n − δn −δγ n + γδ n
" # " #
Xn+1 X1
Since = Xn , we have
Xn X0

γ n+1 − δ n+1 −δγ n+1 + γδ n+1

" # " #" #
Xn+1 −1
X1
= (γ − δ )
Xn γ n − δn −δγ n + γδ n X0

X1 γ n+1 − X1 δ n+1 − X0 δγ n+1 + X0 γδ n+1

" #
−1
= (γ − δ )
X1 γ n − X1 δ n − X0 δγ n + X0 γδ n

Thus
X1 γ n − X1 δ n − X0 δγ n + X0 γδ n
Xn =
γ−δ
1 h
i
X1 − δX0 γ n + γX0 − X1 δ n


=
γ−δ
Let
1
Xn = (V3 + V4 )
γ−δ
where

V3 = X1 − δX0 γ n


h i
= s (1 + i) − t (2 + i) − δ (s − 2t)

= is + s − it − 2t − δs + 2δt γ n

= isγ n + sγ n − sδγ n − itγ n − 2tγ n + 2δtγ n

= isγ n + sγ n − s (2 + i − γ ) γ n − itγ n − 2tγ n + 2t (2 + i − γ ) γ n
By Eqn. (3.7)
298 A. A. Wani, G. C. Morales and N. A. Malik

= isγ n + sγ n − 2sγ n − isγ n + sγ n+1 − itγ n − 2tγ n + 4tγ n + i2tγ n − 2tγ n+1
= −sγ n + sγ n+1 + itγ n + 2tγ n − 2tγ n+1
= sγ n+1 − sγ n + tγ n 2 + i − 2γ

= sγ n+1 − sγ n + tγ n γ + δ − 2γ


By Eqn. (3.7)
= sγ n+1 − sγ n − tγ n (γ − δ )
Similarly
V4 = −sδ n+1 + sδ n − tδ n (γ − δ )
Therefore
1 h n+1 i
Xn = sγ − sγ n − tγ n (γ − δ ) − sδ n+1 + sδ n − tδ n (γ − δ )
γ−δ
1 h n+1 i
= sγ − sδ n+1 − sγ n + sδ n − tγ n (γ − δ ) − tδ n (γ − δ )
γ−δ
!
γ n+1 − δ n+1 γ n − δn  
=s − − t γ n + δn
γ−δ γ−δ

Hence the result.

Again we know that the recurrence relation for hXn i is a second order homogeneous linear
recurrence relation. Then the general solution or nth term of the binomial sequence hXn i is also
given according to
Xn = Aγ n + Bδ n (3.9)
where A and B are constants and the values of A and B are as
X1 − δX0 γX0 − X1 d
A= and B = ⇒ AB =
2 (3.10)
γ−δ γ−δ γ−δ
where d is the fixed quantity dependent only on X0 and X1 .
Now we express the binomial sequence hXn i in terms of two sequences hMn i and hNn i, where
γ n − δn
Mn =
γ−δ
M1 − δM0 γM0 − M1
= A1 γ n + B 1 δ n , A1 = , A2 = (3.11)
γ−δ γ−δ
−1
⇒ A1 A2 =
2
γ−δ
and
Nn = γ n + δ n
N1 − δN0 γN0 − N1 (3.12)
= B1 γ n + B2 δ n , B1 = , B2 = ⇒ B1 B2 = 1
γ−δ γ−δ

E Mn+1 − Mn is the binomial form or binomial sequence of the Jacobsthal-Like sequence

DClearly
Un and hNn i is the binomial form or binomial sequence of the Jacobsthal-Lucas-Like se-
b
quence U n .

4 Matrix Sequence of the Binomial Sequence hXn i

In this section we define a matrix sequence hZn i by using binomial sequence hXn i and so called
hZn i as binomial matrix sequence. In addition to this we give some results related to sequences
hXn i, hMn i, hNn i and hZn i.
 MATRIX SEQUENCE OF THE BINOMIAL FORM 299

√

Definition 4.1. For i = −1 , the binomial matrix sequence hZn in∈N is defined by the follow-
ing equation:

Zn+1 = (2 + i) Zn + (1 − i) Zn−1 , n ≥ 1 (4.1)

" # " #
4 + 3i 3−i 8 + 9i 7 − i
with Z0 = and Z1 =
2+i 1−i 4 + 3i 3 − i

Some few initial few terms of the the binomial matrix sequence hZn in∈N are given by
" # " # " #
4 + 3i 3 − i 8 + 9i 7−i 14 + 25i 17 + i
Z0 = , Z1 = , Z2 =
2+i 1−i 4 + 3i 3 − i 8 + 9i 7−i
" #
20 + 65i 39 + 11i
Z3 =
14 + 25i 17 + i

and so on.
As we know that the elements of the binomial matrix sequence hZn i are in the form of of matrices
and the entries of these matrices are the elements of binomial sequence hXn i. Now in the next
theorem we give the nth term of the binomial matrix sequence hZn i in terms of the binomial
sequence hXn i.

Theorem 4.2. For n ∈ Z0 , the nth term of the matrix sequence hZn i is given by



X0 Xn+4 − X1 Xn+3 1 − i X0 Xn+3 − X1 Xn+2
Zn = d−1 


 (4.2)
X0 Xn+3 − X1 Xn+2 1 − i X0 Xn+2 − X1 Xn+1
" #
2+i 1−i
Proof. Let Z = be a square matrix correspond to the binomial matrix sequence
1 0
" # " #
Zn+1 Z1
hZn i and assuredly = Zn . Then by similar manner from the proof of the Theorem
Zn Z0
(3.4), we write
Z1 γ n − Z1 δ n − Z0 δγ n + Z0 γδ n
Zn =
γ−δ
1 h
i
Z1 − δZ0 γ n + γZ0 − Z1 δ n


=
γ−δ
" ! ! !
1 8 + 9i 7 − i n 4 + 3i 3 − i 4 + 3i 3 − i
= γ − δγ n + γδ n
γ−δ 4 + 3i 3 − i 2+i 1−i 2+i 1−i
! #
8 + 9i 7 − i n
− δ (4.3)
4 + 3i 3 − i

" ! ! !
AB γ − δ 8 + 9i 7 − i 4 + 3i 3 − i 4 + 3i 3 − i
= γn − δγ n + γδ n
d 4 + 3i 3 − i 2+i 1−i 2+i 1−i
! #
8 + 9i 7 − i n
− δ By Eqn. (3.10)
4 + 3i 3 − i

" #
AB γ − δ a1 a2
=
d a3 a4
300 A. A. Wani, G. C. Morales and N. A. Malik

Here
a1 = 8 + 9i γ n − 4 + 3i δγ n + 4 + 3i γδ n − 8 + 9i δ n

= 8γ n + 9γ n i − 4γ n 2 + i − γ − 3γ n i 2 + i − γ + 4δ n 2 + i − γ

+ 3δ n i 2 + i − γ − 8δ n − 9δ n i

= 3γ n − γ n i + 4γ n+1 + 3γ n+1 i − 3δ n + δ n i − 4δ n+1 − 3δ n − 3δ n+1 i

h
i h
i
= γ n 4 + 3i γ + 3 − i − δ n 4 + 3i δ + 3 − i

Since

2

2

4 + 3i = 2 + i + 1 − i = γ + δ + γδ and





3 − i = 2 + i 1 − i = − γ + δ γδ
Hence, we get
h
2
i h
2
i
a1 = γ n γ + δ γ − γδ γ − γ + δ γδ − δ n γ + δ δ − γδ δ − γ + δ γδ

= γ n+3 − δ n+3
Therefore
AB γ − δ γ n+3 − AB γ − δ δ n+3




AB γ − δ
a1 =
d d

n+3
− B γX1 − δX0 δ n+3


A γX0 − X1 γ
= By Eqn. (3.10)
d
n+4 n+4
− X1 Aγ n+3 + Bδ n+3



X0 Aγ + Bδ
=
d
X0 Xn+4 − X1 Xn+3
=
d
Now
a2 = 7 − i γ n − 3 − i δγ n + 3 − i γδ n − 7 − i δ n

= 7γ n − iγ n − 3γ n 2 + i − γ + γ n i 2 + i − γ + 3δ n 2 + i − δ − δ n i 2 + i − δ

− 7δ n + δ n i
= γ n −2i + γ 3 − i + δ n 2i − δ 3 − i

 


Since





−2i = 1 − i 1 − i = γδ γδ and





3 − i = 2 + i 1 − i = − γ + δ γδ
This implies that

h
i
h
i
a2 = 1 − i γ n −γδ + γ γ + δ + 1 − i δ n γδ − δ γ + δ

 
= 1 − i γ n+2 − δ n+2

Therefore

AB γ − δ γ n+2 − AB γ − δ δ n+2




AB γ − δ
a2 = 1 − i
d d

n+2
− B γX1 − δX0 δ n+2



A γX0 − X1 γ
= 1−i
d
 MATRIX SEQUENCE OF THE BINOMIAL FORM 301

By Eqn. (3.10)

X0 Aγ n+3 + Bδ n+3
− X1 Aγ n+2 + Bδ n+2



= 1−i
d

X0 Xn+3 − X1 Xn+2
= 1−i
d
Similarly


AB γ − δ X0 Xn+3 − X1 Xn+2
a3 = and
d d


AB γ − δ
X0 Xn+2 − X1 Xn+1
a4 = 1 − i
d d
Thus, we get



X0 Xn+4 − X1 Xn+3 1 − i X0 Xn+3 − X1 Xn+2
Zn = d−1 



X0 Xn+3 − X1 Xn+2 1 − i X0 Xn+2 − X1 Xn+1

Lemma 4.3. For n ∈ Z0 , we have

X0 Xn+1 − X1 Xn
Mn = (4.4)
d
Theorem 4.4. For n ∈ Z0 , the following result holds
"
#
Mn+3 1 − i Mn+2
Zn =
(4.5)
Mn+2 1 − i Mn+1

Proof. The proof of this theorem is clearly visible from the equations (4.2) and (4.4).
Theorem 4.5. For n ∈ Z0 , we have




−2 N0 Nn+4 − N1 Xn+3 1 − i N0 Nn+3 − N1 Nn+2
Zn = γ − δ 



 (4.6)
N0 Nn+3 − N1 Nn+2 1 − i N0 Nn+2 − N1 Nn+1

Proof. By using Equation (4.3) from the proof of Theorem (4.2), we have
" ! ! !
1 8 + 9i 7 − i n 4 + 3i 3 − i 4 + 3 i 3 − i
Zn = γ − δγ n + γδ n
γ−δ 4 + 3i 3 − i 2+i 1−i 2+i 1−i
! #
8 + 9i 7 − i n
− δ
4 + 3i 3 − i
" #
1 a1 a2
=
γ − δ a3 a4

Since

a1 = γ n+3 − δ n+3

We have
γ − δ γ n+3 − δ n+3



a1
=
2
γ−δ γ−δ
302 A. A. Wani, G. C. Morales and N. A. Malik

B1 B2 γ − δ γ n+3 − δ n+3



=
2 By Eqn. (3.12)
γ−δ
B1 B2 γ − δ γ n+3 − B1 B2 γ − δ δ n+3



=
2
γ−δ
B1 γN0 − N1 γ n+3 − B2 N1 − δN0 δ n+3



=
2 By Eqn. (3.12)
γ−δ
   
N0 B1 γ n+4 + B2 δ n+4 − N1 B1 γ n+3 + B2 δ n+3
=
2
γ−δ
N0 Nn+4 − N1 Nn+3
=
2
γ−δ

Again

γ − δ γ n+2 − δ n+2



a2
= 1−i
2
γ−δ γ−δ

B1 B2 γ − δ γ n+2 − δ n+2



= 1−i
2 By Eqn. (3.12)
γ−δ

B1 B2 γ − δ γ n+2 − B1 B2 γ − δ δ n+2



= 1−i
2
γ−δ

B1 γN0 − N1 γ n+2 − B2 N1 − δN0 δ n+2



= 1−i
2 By Eqn. (3.12)
γ−δ
   
n+3

N0 B 1 γ + B2 δ n+3 − N1 B1 γ n+2 + B2 δ n+2
= 1−i
2
γ−δ

N0 Nn+3 − N1 Nn+2
= 1−i
2
γ−δ

Equivalently

a3 N0 Nn+3 − N1 Nn+2
=
2 and
γ−δ γ−δ

a4
N0 Nn+2 − N1 Nn+1
= 1−i
2
γ−δ γ−δ

Hence, we achieve




−2 N0 Nn+4 − N1 Xn+3 1 − i N0 Nn+3 − N1 Nn+2
Zn = γ − δ 




N0 Nn+3 − N1 Nn+2 1 − i N0 Nn+2 − N1 Nn+1

Corollary 4.6. For n ∈ Z0 , the ensuing results hold

X0 Xn+2 − X1 Xn+1 = dMn+1 (4.7)

2
N0 Nn+2 − N1 Nn+1 = γ − δ Mn+1 (4.8)
 MATRIX SEQUENCE OF THE BINOMIAL FORM 303

Corollary 4.7. Let n ≥ 0, the following properties hold

X0 Xn+4 + Xn+2 − X1 Xn+3 + Xn+1 = dNn+2 (4.9)



2
N0 Nn+4 + Nn+2 − N1 Nn+3 + Nn+1 = γ − δ Nn+2 (4.10)

Proof. By equating corresponding terms of matrices from the Equations (4.2) and (4.5), we have

X0 Xn+4 − X1 Xn+3 = dMn+3

X0 Xn+2 − X1 Xn+1 = dMn+1

Adding together both the equations, we get

X0 Xn+4 + Xn+2 − X1 Xn+3 + Xn+1


= d Mn+3 + Mn+1
d  n+3 
= γ − δ n+3 + γ n+1 − δ n+1 By Eqn. (3.11)
γ−δ
d h n+1 2
i
γ + 1 − δ n+1 δ 2 + 1


= γ
γ−δ
d h n+2 i
γ − δ + δ n+2 γ − δ )


= γ By Eqn. (3.7)
γ−δ
= dNn+2 By Eqn. (3.12)

Hence the result.

Conclusion
In this paper we studied the matrix sequence of the binomial form of second order Jacobsthal-
Like sequence. In addition to this we obtained some basic results about the said matrix sequence.
As an extension of this article, future work will examine the matrix sequence of the binomial
form of other second order sequences or higher order sequences.

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304 A. A. Wani, G. C. Morales and N. A. Malik

Author information
A. A. Wani, University of Kashmir, J and K India.
E-mail: arfatmaths@gmail.com
G. C. Morales, Departamento de Matemticas,Universidad Tecnica Federico, Santa Mara, Valparaso, Chile.
E-mail: gamaliel.cerda.m@mail.pucv.cl
N. A. Malik, Cluster University of Jammu, J and K India.
E-mail: drnaseerulhassan@gmail.com

Received: July 29, 2020.

Accepted: November 22, 2020