International Journal of Analysis and Applications

Volume 16, Number 6 (2018), 894-903

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-894

ON INTEGRATED AND DIFFERENTIATED C2-SEQUENCE SPACES

LAKSHMI NARAYAN MISHRA1,2,∗, SUKHDEV SINGH3, VISHNU NARAYAN MISHRA4

1Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, University,

Vellore 632014, TN, India

2L. 1627 Awadh Puri Colony Beniganj, Phase -III, Opposite - Industrial Training Institute (I.T.I.),

Ayodhya Main Road Faizabad-224 001, UP, India

3Department of Mathematics, Lovely Professional University, Jalandhar-Delhi Road, Phagwara-144411,

Punjab, India

4Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur,

Madhya Pradesh 484887, India

∗Corresponding author: lakshminarayanmishra04@gmail.com

Abstract. The integrated and differentiated C2-sequence spaces are defined and studied by using the norm

on the bicomplex space C2, infinite matrices of the bicomplex number and the Orlicz functions. We also

studied some topological properties of the C2-sequence spaces We define the α-duals of the integrated and

differentiated C2-sequence spaces.

Received 2018-02-28; accepted 2018-05-24; published 2018-11-02.

2010 Mathematics Subject Classification. 06F30, 54A99.

Key words and phrases. Bicomplex numbers; bicomplex net; C2-sequence spaces.
c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

894

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-894


Int. J. Anal. Appl. 16 (6) (2018) 895

1. Introduction

The set of bicomplex numbers [8] is denoted by C2 and sets of real and complex numbers are denoted as

C0 and C1, respectively.The set of bicomplex number is defined as (cf. [8], [9])

C2 := {a1 + i1a2 + i2a3 + i1i2a4 : ak ∈ C0, 1 ≤ k ≤ 4}

:= {w1 + i2w2 : w1,w2 ∈ C1}

where i21 = i
2
2 = −1, i1i2 = i2i1.

The set of bicomplex numbers C2 have exactly two non-trivial idempotent elements denoted by e1 and

e2 give as e1 = (1 + i1i2)/2 and e2 = (1 − i1i2)/2. Note that e1 + e2 = 1 and e1.e2 = 0. The number

ξ = w1 + i2w2 can be uniquely expressed as a complex combination of e1 and e2 [8].

ξ = w1 + i2w2 =
1ξe1 +

2ξe2, (1.1)

where 1ξ = w1 − i1w2 and 2ξ = w1 + i1w2. The complex coefficients 1ξ and 2ξ are called the idempotent

components of ξ, and 1ξe1 +
2ξe2 is known as idempotent representation of bicomplex number ξ.

The auxiliary complex spaces A1 and A2 are defined as follows:

A1 =
{

1ξ : ξ ∈ C2
}

and A2 =
{

2ξ : ξ ∈ C2
}
.

The norm in C2 is defined as follows:

||ξ|| =
√
a21 + a

2
2 + a

2
3 + a

2
4 =

√
|w1|2 + |w2|2 =

√
|1ξ|2 + |2ξ|2

2
(1.2)

Further, the norm of the product of two bicomplex numbers and the product of their norms are connected

by means of the following inequality:

||ξ .η|| ≤
√

2 ||ξ|| . ||η|| (1.3)

The inequality given in (1.3) is the best possible relation . For this reason, we call C2 as modified complex

Banach algebra [8].

Throughout the paper, the ω4, c, c0 and `
∞
C2 denote the space of all bicomplex sequences, convergent

sequences, null sequences and all bounded sequences. We denote the zero sequence (0, 0, 0, . . . , 0, . . .) by π.

Refer the book by Mursaleen [?] for details about summability methods.

The Orlicz function M is defined as M : [0,∞) → [0,∞). It is continuous, non-decreasing and M(0) =

0,M(x) > 0 for x > 0. Also, for λ ∈ (0, 1) it satisfies the condition

M(λx + (1 −λ)y) ≤ λM(x) + (1 −λ)M(y) (1.4)

and If the condition of convexity of the Orlicz function M is replaced by M(x + y) ≤M(x) + M(y), then

the function M is called the modulus function.



Int. J. Anal. Appl. 16 (6) (2018) 896

The notations (X : Y ) denote the class of all matrices M, such that M : X → Y . Therefore, M ∈ (X : Y )

if and only if M(x) = {(Mx)n}n∈N ∈ Y .

A sequence {ξn} in C2 is said to be M-summable to the bicomplex number ξ if M(ξn) converges to ξ

which is called M-limit of {ξn}.

In [1], the sequence space bvp is defined, which have all sequences such that their ∆-transform is in `p,

where ∆ denotes the matrix ∆ = {δnm} as

δnm :=




(−1)n−k , n− 1 ≤ m ≤ n

0 , 0 ≤ m ≤ n− 1 or k > n
(1.5)

We consider following matrices for our C2-sequence spaces.

ωnm :=



ξ , 1 ≤ m ≤ nξ

0 , ξ ≺Id η
(1.6)

γnm :=



ξ , m = n

−ξ , n− 1 = m

0 , otherwise

(1.7)

πnm :=



ξ−1 , 1 ≤ m ≤ n

0 , m > n

(1.8)

and

πnm :=



ξ−1 , n = m

−ξ−1 , n− 1 = m

0 otherwise

(1.9)

Here we must note that ξ−1 exists if and only if ξ ∈ C2/O2.

The integrated and differentiated sequence space were first studied by Goes and Goes [3]. In this paper,

we define and study some C2-sequence space. In the last section we studied the α-dual of these sequence

spaces.

2. Bicomplex integrated (int) and differentiated (diff ) C2-sequence spaces

Goes and Goes [3] has given the concept of the integrated sequence space. In this section we will obtain the

matrix domains of the sequence space `1 by using the bicomplex matrices. We shall show that the integrated

and differentiated C2-sequence spaces are Banach Spaces, BK-spaces, norm isomorphic to `1, separable these



Int. J. Anal. Appl. 16 (6) (2018) 897

spaces have AK-property. The spaces
∫
bv and

∫
`1 have monotone norms and therefore the spaces

∫
bv and∫

`1 have AK-property. Let ω4 denote the family of bicomplex sequences.

Now we are giving the definitions of some C2-sequence spaces as follows:

Definition 2.1 (Integrated C2-sequence spaces).

`1(C2,M,‖.‖) =
{
{ξn}∈ ω4 :

∞∑
n=1

M
(
‖nξn‖
K

)
< ∞, for some K > 0

}
and

bv(C2,M,‖.‖) :=

{
ξ = {ξn}∈ ω4 :

∞∑
n=2

M
(
‖∆(nξn)‖

K

)
< ∞, for some K > 0

}

Definition 2.2 (Differentiated C2-sequence spaces).

`1(C2,M,‖.‖) :=

{
ξ = {ξn}∈ ω4 :

∞∑
n=1

M
(
‖ξn/n‖
K

)
< ∞, for some K > 0

}

bv(C2,M,‖.‖) :=

{
ξ = {ξn}∈ ω4 :

∞∑
n=2

M
(
‖∆(ξn/n)‖

K

)
< ∞, for some K > 0

}

we can redefine the spaces `(C2,M,‖.‖), bv(C2,M,‖.‖), `(C2,M,‖.‖) and bv(C2,M,‖.‖) by

(`1)Ω = `1(C2,M,‖.‖), (`1)Γ = bv(C2,M,‖.‖), (`1)Π = `1(C2,M,‖.‖), (`1)λ = bv(C2,M,‖.‖).

Let ξ = {ξn}∈ `1(C2,M,‖.‖). Then the Ω-transform of ξ is defined as

ζn := (Ω(ξ))n =

n∑
m=1

M
(
‖(mξm)‖

K

)
for some K > 0

or equivalently,

1ζn := (Ω(
1ξ))n =

n∑
m=1

M
(
|m 1ξm|
K

)
and 2ζn := (Ω(

2ξ))n =

n∑
m=1

M
(
|m 2ξm|
K

)
Let ξ = {ξn}∈ bv(C2,M,‖.‖). The Γ-transform of {ξn} is defined as

ζn := (Ω(ξ))n =



ξ1 , p = 1

∆(pξp) , p ≥ 2

Let ξ = {ξn}∈ `1(C2,M,‖.‖). The Π-transform of {ξn} is defined as

ζn = (Π ξ)n =

n∑
p=1

M
(
‖ξp/p‖
K

)
for some K > 0

Let ξ = {ξn}∈ bv(C2,M,‖.‖). The Σ-transform of {ξn} is defined as



Int. J. Anal. Appl. 16 (6) (2018) 898

ζn := (Σ(ξ))n =



ξ1 , p = 1

∆(p−1 ξp) , p ≥ 2

For the convenience, we use the following notations.

K1 = `1(C2,M,‖.‖), K2 = bv(C2,M,‖.‖), K3 = `1(C2,M,‖.‖), K4 = bv(C2,M,‖.‖).

Proposition 2.1. A sequence {ξn} is in X(C2,M,‖.‖) if and only if {1ξn} ∈ S(A1,M,‖.‖) and {2ξn} ∈

S(A2,M,‖.‖), where X = K1,K2,K3 and K4.

Theorem 2.1. The space `1(C2,M,‖.‖) is a linear space over C0.

Proof. Let {ξn},{ηn}∈ `1(C2,M,‖.‖). Then there exist P1 > 0 and P2 > 0 such that
∞∑
n=1

M
(
‖nξn‖
P1

)
< ∞ and

∞∑
n=1

M
(
‖nξn‖
P2

)
< ∞

Now let α,β ∈ C2 \ O2 and P = max{2‖α‖P1, 2‖β‖P2}. Then
∞∑
k=1

M
(
‖α∆(k ξk) + β∆(k ηk)‖

P

)
≤
∞∑
k=1

M
(
‖α ∆(k ξk)‖

P1

)
+

∞∑
k=1

M
(
‖β ∆(k ηk)‖

P2

)
.

Therefore, {αξn +β ηn}∈ `1(C2,M,‖.‖). Hence, the space `1(C2,M,‖.‖) is a linear space over C2\O2. �

Lemma 2.1. The functions ‖ξ‖`1(C2,M,‖.‖) =
∑∞
m=1 ‖ωnmξm‖ and ‖ξ‖`1(C2,M,‖.‖) =

∑∞
m=1 ‖πnmξm‖ are

norms on the spaces `1(C2,M,‖.‖) and `1(C2,M,‖.‖), respectively.

Theorem 2.2. The spaces `1(C2,M,‖.‖) and `1(C2,M,‖.‖) are Banach spaces with norms ‖ξ‖`1(C2,M,‖.‖) =∑∞
m=1 ‖ωnmξm‖ and ‖ξ‖`1(C2,M,‖.‖) =

∑∞
m=1 ‖πnmξm‖, respectively.

Proof. Let {ξnk} be a Cauchy sequence in `1(C2,M,‖.‖). Then for given � > 0, ∃ m0 ∈ N such that

‖ξnk − ξ
m
k ‖ < �, ∀n,m > m0 (2.1)

Therefore,

∑
k

‖Ω(ξm)k − Ω(ξn)k‖ < �, ∀n,m > m0

⇒ {Ω(ξ1)k, Ω(ξ2)k, Ω(ξ3)k, . . . , Ω(ξn)k, . . .} is a Cauchy Sequence of bicomplex numbers. Since, C2 is a

modified Banach space. Therefore, {Ω(ξn)k} is convergence in C2. Suppose that

Ω(ξn)k → Ω(ξ), n →∞,∀k

Using all these limits, we define a sequence {Ω(ξ)1, Ω(ξ)2, Ω(ξ)3, . . . ,}.



Int. J. Anal. Appl. 16 (6) (2018) 899

and from equation (2.1), we have

p∑
k=1

‖Ω(ξm)k − Ω(ξn)k‖ < � (2.2)

For any n > m0, by letting m →∞ and p →∞, we have

‖ξn − ξ‖`1(C2,M,‖.‖) ≤ �

In particular,

‖ξ‖`1(C2,M,‖.‖) ≤ K + ‖ξ
n‖`1(C2,M,‖.‖), for some K ≥ �.

Hence, ξ ∈ `1(C2,M,‖.‖). Further, ξn → ξ. Therefore, `1(C2,M,‖.‖) is complete. �

Corollary 2.1. The space `1(C2,M,‖.‖) is a Banach space.

Theorem 2.3. The spaces `1(C2,M,‖.‖) and `1(C2,M,‖.‖) are BK-spaces with norms ‖ξ‖`1(C2,M,‖.‖) =∑∞
m=1 ‖ωnmξm‖ and ‖ξ‖`1(C2,M,‖.‖) =

∑∞
m=1 ‖πnmξm‖, respectively.

Proof. Let {ξn}∈ `1(C2,M,‖.‖). Define fp(ξn) = ξp,∀n ∈ N. Then

‖ξn‖`1(C2,M,‖.‖) =
∑
‖nξn‖

So that ‖nξn‖≤‖ξn‖`1(C2,M,‖.‖) ⇒‖ξn‖≤ K‖ξn‖`1(C2,M,‖.‖) ⇒‖fn(ξp)‖≤ K‖ξn‖`1(C2,M,‖.‖).

Therefore, fn is a continuous linear functional for each n. So, `1(C2,M,‖.‖) is a BK-space. �

In the similar manner, we can prove that `1(C2,M,‖.‖) is a BK-space.

Theorem 2.4. The space bv(C2,M,‖.‖) is a BK-space with the norm ‖ξ‖bv(C2,M,‖.‖) =
∑∞
m=1 ‖∆(mξm)‖.

Proof. As we know, bv(C2,M,‖.‖) = (`1)Σ is true and `1 is a BK-space with respect to the norm ‖ξ‖`1 and

also the matrix Σ is a triangular matrix.Then by Wilansky [?], the space bv is a BK-space. �

Theorem 2.5. The function ‖ξ‖bv(C2,M,‖.‖) =
∑∞
m=1 ‖∆(mξm)‖ is a norm on bv(C2,M,‖.‖).

Theorem 2.6. The spaces bv(C2,M,‖.‖) and bv have AK-property.

Proof. Let {ξnk}∈ bv(C2,M,‖.‖) and [ξ
n
k ] = {ξ

n
1 ,ξ

n
2 ,ξ

n
3 , . . . , . . . ,ξ

n
k , 0, 0, 0, . . .}.

ξnk − [ξ
n
k ] = {0, 0, 0, . . . ,ξ

n
k+1,ξ

n
k+2, . . . ,}.

⇒ ‖ξnk − [ξ
n
k ]‖bv(C2,M,‖.‖) = ‖0, 0, 0, . . . ,ξ

n
k+1,ξ

n
k+2, . . . ,‖bv(C2,M,‖.‖).

=
∑
p≥k+1

M
(
‖ξnp /p‖
K

)
→ 0, as p → 0.

⇒ [ξnk ] → ξ
n
k as k →∞



Int. J. Anal. Appl. 16 (6) (2018) 900

Then, the space bv(C2,M,‖.‖) has AK-property. �

Theorem 2.7. The spaces `1(C2,M,‖.‖), bv(C2,M,‖.‖), `1(C2,M,‖.‖) and bv(C2,M,‖.‖) are norm iso-

morphic to `1.

Proof. We must show that there is a one-one and onto linear mapping between bv(C2,M,‖.‖) and `1.

Suppose that T : bv(C2,M,‖.‖) → `l be a mapping defined as ξ 7→ Tξ.

Clearly, for ξ = θ ⇒ Tξ = θ.

Now, let η ∈ `1. Define a sequence {ξk}∈ bv(C2,M,‖.‖) by

ξk =
1

k

k∑
p=1

yp

Then

‖ξk‖bv(C2,M,‖.‖) =
∑
k

∆(k ξk) =
∑
k

∥∥∥∥ k∑
p=1

pηp − (p− 1)
k−1∑
p=1

ηp

∥∥∥∥
=

∑
k

‖ηk‖ = ‖η‖`1

Therefore, ξn ∈ bv(C2,M,‖.‖). Hence, the spaces bv(C2,M,‖.‖) and `1 are isomorphic. �

In the similar way, we can prove the isomorphism of remaining spaces.

Theorem 2.8. The spaces `1(C2,M,‖.‖) and bv(C2,M,‖.‖) have monotone norm.

Proof. Let {ξn}∈ bv(C2,M,‖.‖).

Define ‖ξn‖bv(C2,M,‖.‖) =
∑
k=1 ∆(kξk)

and ‖[ξp]‖bv(C2,M,‖.‖) =
∑n
k=1 ‖∆(pξp)‖, ∀{ξk}∈ bv(C2,M,‖.‖).

Now, suppose q > p, then

‖[ξp]‖bv(C2,M,‖.‖) =
p∑
k=1

‖∆(k ξk)‖

≤
q∑

k=1

‖∆(k ξk)‖

≤ ‖[ξq]‖bv(C2,M,‖.‖)

Also,

sup‖[ξn]‖bv(C2,M,‖.‖) = sup
( n∑
k=1

‖∆(k ξk)‖
)

= ‖ξn‖bv(C2,M,‖.‖).

Therefore, the space bv(C2,M,‖.‖) has the monotone norm. �



Int. J. Anal. Appl. 16 (6) (2018) 901

Remark 2.1. The spaces `1 and bv(C2,M,‖.‖) have AB-property.

Theorem 2.9. The following statements hold for bv(C2,M,‖.‖) and bv(C2,M,‖.‖) given as :

(1) If ζ(m) = {ζ(m)n } is sequence where {ζ
(m)
n }∈ bv(C2,M,‖.‖) of elements of bv(C2,M,‖.‖), defined as

ζ(m)n :=




1/m , n ≥ m

0 , n < m

This sequence is the basis for the space bv(C2,M,‖.‖) and select Bm = (Mξ)m, for all m ∈ N and

matrix M defined in equation (??), then ξ ∈ bv(C2,M,‖.‖) has the unique representation of the

type:

ξ =
∑
m

(Mξ)m ζ
(m)
n

(2) Define a sequence {ηmn } with ηmn ∈ bv(C2,M,‖.‖) as

η(m)n :=



m , n ≥ m

0 , n < m

Then this sequence ζ(m) is a basis for the space bv and for Em = (Ax)m, for all m ∈ N, where

the matrix A is defined by Γ = [γnm], every sequence ξ ∈ bv have unique representation as

ξ =
∑
m

Emζ
(m)

Corollary 2.2. The spaces bv(C2,M,‖.‖) and bv(C2,M,‖.‖) are separable.

3. α−Duals of the C2-Sequence Spaces

In this section, we determine the α−duals of the spaces K2 and K4.

Let ξ = {ξn} and η = {ηn} be sequences, and A and B be two subsets of ω4. Now let M = (amk) be an

infinite matrix of bicomplex numbers. Define ξη = (ξnηn),

ξ−1n ? B = {ζ ∈ ω4 : ζ ξ ∈ B}. N(A,B) = ∩ξ∈Aξ−1 ? B = {ζ ∈ ω4 : ζ ξ ∈ B, for ξ ∈ A}. In particular, for

B = `1,cs or bs, We have ξ
α = ξ−1 ? `1, ξ

β = ξ1 ? cs and ξγ = ξ−1 ? bs. The α− dual of A are given by

Aα = M(A,`1).

Suppose that Mm = (amk)
∞
k=0 denotes the m-th row of the matrix M. Let Mm(ξ) =

∑∞
k=0 amkξk,

∀n = 0, 1, 2, . . ., and M(ξ) = [Mm(ξ)]∞m=0, where Mm ∈ ξβ

Lemma 3.1. [?] Let A1,A2 be to BK-spaces, and M = [ηnm] be a triangular matrix where ξnm ∈ C2/O2,

then for matrix SMA1 = [ξnm] defined with ν = {νm}∈ A1 as



Int. J. Anal. Appl. 16 (6) (2018) 902

ξnm =

n∑
i=1

νi ηnm µim

Then A2A1(M) ⊂ A1(M) holds if and only if the matrix SMA1 = MDνM
−1 ∈ (A1 : A1), where Dν is a

diagonal matrix such that [Dν]nn = νn, ∀n ∈ N.

Lemma 3.2. [?] Let {γk} be a sequence in ω4 and M = [ηnm] be an invertible triangular matrix. Define a

matrix SMA1 = [ξnm] as

ξnm =

n∑
i=m

ηi µim

Then

A
β
1 (M) = {ηm ∈ ω4 : S(M) ∈ (A1 : c)}

and

A
γ
1 (M) = {ηm ∈ ω4 : S(M) ∈ (A1 : `∞)}

Lemma 3.3. Let M = [ξnm] be an infinite matrix of bicomplex numbers. Then

(1) M ∈ (`1 : `1) ⇐⇒ sup
∑
k∈N ‖ξnm‖ < ∞.

(2) M ∈ (`1 : `∞) ⇐⇒ supk,n∈N ‖ξnm‖ < ∞

(3) M ∈ (`1,c) ⇐⇒ supk,n∈N ‖ξnm‖ < ∞ and for some sequence {κm} such that lim
n→∞

ξnm = κm

Theorem 3.1. For the space bv(C2,M,‖.‖), we have

bv(C2,M,‖.‖)α = α1

where

α1 =

{
ξ = {ξn}∈ ω4 :

∑
k

∥∥∥∥ n∑
m=1

M
(
‖∆(ξm/m)‖

K

)
ηk

∥∥∥∥ < ∞, (ηk) ∈ bv(C2,M,‖.‖) for some K > 0
}

Proof. {ξn} be any sequence in ω4. Assume the following relation

ξnηn =

n∑
k=1

M
(
‖∆(ξn/n)‖

K

)
ηk = (Eη)k

where E = {enk} is defined by

enm =



M
(
‖∆(ξn/n)‖

K

)
, 1 ≤ m ≤ n

0 , n < m

(3.1)

Therefore, from the equation (3.1) and the Lemma (3.3) we have{
M
(
‖∆(ξn/n)‖

K

)
ζn

}
∈ `1 if and only if Eη ∈ `1, whenever η ∈ `1.

So, ξ = {ξns}∈ bv(C2,M,‖.‖)α if and only if E ∈ (bv(C2,M,‖.‖) : `1).

Hence proved. �



Int. J. Anal. Appl. 16 (6) (2018) 903

Analogously, we can prove the following theorems.

Theorem 3.2. For the space bv(C2,M,‖.‖)

bv(C2,M,‖.‖)α = α2

where

α1 =

{
ξ = {ξn}∈ ω4 :

∑
k

∥∥∥∥ n∑
m=1

M
(
‖(ξm/m)‖

K

)
ηk

∥∥∥∥ < ∞, (ηk) ∈ bv(C2,M,‖.‖) for some K > 0
}

Theorem 3.3. For the space `1(C2,M,‖.‖)

`1(C2,M,‖.‖)α = α1

where

α1 =

{
ξ = {ξn}∈ ω4 :

∑
k

∥∥∥∥ n∑
m=1

M
(
‖(mξm)‖

K

)
ηk

∥∥∥∥ < ∞, (ηk) ∈ bv(C2,M,‖.‖) for some K > 0
}

Theorem 3.4. For the space `1(C2,M,‖.‖)

`1(C2,M,‖.‖)α = α2

where

α1 =

{
ξ = {ξn}∈ ω4 :

∑
k

∥∥∥∥ n∑
m=1

M
(
‖(∆(mξm))‖

K

)
ηk

∥∥∥∥ < ∞, (ηk) ∈ bv(C2,M,‖.‖) for some K > 0
}

References

[1] B. Atlay, F. Basar, Summability Theory and its Applications, Bentham Science Publishers, e-books, Monographs, Istambul,

(2012).

[2] B. Atlay, F. Basar, Certain topological properties and duals of the domain of a triangle matrix in a sequence spaces, J.

Math. Anal. Appl., 336 (2007), 632-645.

[3] Goes and Goes, Sequences of bounded variation and sequences of Fourier coefficients I, Math. Z, 118 (1970) 93-102.

[4] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24(2),169-176, (1981).

[5] M. Et, R. Colak, On some generalized difference sequence spaces, Soochow J. Math., 21(4), 377-386, (1995).

[6] H. Nakano, Modular Sequence Spaces, Proc. Japan Acad., 27, 508-512, (1951).

[7] S.D. Prashar and B Chaudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Mathematics, 25,

419-478, (1994).

[8] G.B. Price, An introduction to multicomplex space and functions, Marcel Dekker, Inc., New York, (1991).

[9] Rajiv K Srivastava, Bicomplex numbers: analysis and applications, Math. Student, 72 (1-4), 63-87, (2007).


	1. Introduction
	2. Bicomplex integrated (int) and differentiated (diff) C2-sequence spaces
	3. -Duals of the C2-Sequence Spaces
	References