International Journal of Analysis and Applications

Volume 16, Number 1 (2018), 16-24

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

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

GENERALIZED ROUGH CESÀRO AND LACUNARY STATISTICAL TRIPLE

DIFFERENCE SEQUENCE SPACES IN PROBABILITY OF FRACTIONAL ORDER

DEFINED BY MUSIELAK-ORLICZ FUNCTION

A. ESI1 AND N. SUBRAMANIAN2,∗

1Department of Mathematics,Adiyaman University,02040,Adiyaman, Turkey

2Department of Mathematics, SASTRA University, Thanjavur-613 401, India

∗Corresponding author: nsmaths@gmail.com

Abstract. We generalized the concepts in probability of rough Cesàro and lacunary statistical by intro-

ducing the difference operator ∆αγ of fractional order, where α is a proper fraction and γ = (γmnk) is any

fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving

lacunary sequence θ and arbitrary sequence p = (prst) of strictly positive real numbers and investigate the

topological structures of related with triple difference sequence spaces.

The main focus of the present paper is to generalized rough Cesàro and lacunary statistical of triple

difference sequence spaces and investigate their topological structures as well as some inclusion concerning

the operator ∆αγ .

1. Introduction

A triple sequence (real or complex) can be defined as a function x : N×N×N → R (C) , where N,R and

C denote the set of natural numbers, real numbers and complex numbers respectively. The different types

of notions of triple sequence was introduced and investigated at the initial by Sahiner et al. [10,11], Esi et

Received 31st August, 2017; accepted 28th November, 2017; published 3rd January, 2018.

2010 Mathematics Subject Classification. 40F05, 40J05, 40G05.

Key words and phrases. analytic sequence; Musielak-Orlicz function; triple sequences; chi sequence; Cesàro summable;

lacunary statistical.

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.

16

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


Int. J. Anal. Appl. 16 (1) (2018) 17

al. [1-3], Datta et al. [4],Subramanian et al. [12], Debnath et al. [5] and many others.

A triple sequence x = (xmnk) is said to be triple analytic if

supm,n,k |xmnk|
1

m+n+k < ∞.

The space of all triple analytic sequences are usually denoted by Λ3. A triple sequence x = (xmnk) is called

triple gai sequence if

((m + n + k)! |xmnk|)
1

m+n+k → 0 as m,n,k →∞.

The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [6] as follows

Z (∆) = {x = (xk) ∈ w : (∆xk) ∈ Z}

for Z = c,c0 and `∞, where ∆xk = xk −xk+1 for all k ∈ N.

The difference triple sequence space was introduced by Debnath et al. (see [5]) and is defined as

∆xmnk = xmnk −xm,n+1,k −xm,n,k+1 + xm,n+1,k+1 −xm+1,n,k + xm+1,n+1,k + xm+1,n,k+1 −xm+1,n+1,k+1

and ∆0xmnk = 〈xmnk〉 .

2. Some new difference triple sequence spaces with fractional order

Let Γ (α) denote the Euler gamma function of a real number α. Using the definition Γ (α) with α /∈

{0,−1,−2,−3, · · ·} cab be expressed as an improper integral as follows: Γ (α) =
∫∞

0
e−xxα−1dx, where α

is a positive proper fraction. We have defined the generalized fractional triple sequence spaces of difference

operator

∆αγ (xmnk) =

∞∑
u=0

∞∑
v=0

∞∑
w=0

(−1)u+v+w Γ (α + 1)
(u + v + w)!Γ (α− (u + v + w) + 1)

xm+u,n+v,k+w. (2.1)

In particular, we have

(i) ∆
1
2 (xmnk) = xmnk − 116xm+1,n+1,k+1 −··· .

(ii) ∆−
1
2 (xmnk) = xmnk +

5
16
xm+1,n+1,k+1 + · · · .

(iii) ∆
2
3 (xmnk) = xmnk − 481xm+1,n+1,k+1 − ··· . Now we determine the new classes of triple difference

sequence spaces ∆αγ (x) as follows:

∆αγ (x) =
{
x : (xmnk) ∈ w3 :

(
∆αγx

)
∈ X

}
, (2.2)

where ∆αγ (xmnk) =
∑∞
u=0

∑∞
v=0

∑∞
w=0

(−1)u+v+wΓ(α+1)
(u+v+w)!Γ(α−(u+v+w)+1)xm+u,n+v,k+w and

X ∈ χ3∆f (x) = χ
3
f

(
∆αγxmnk

)
= µmnk

(
∆αγx

)
=
[
fmnk

((
(m + n + k)!

∣∣∆αγ∣∣) 1m+n+k , 0̄)] .
Proposition 2.1. (i) For a proper fraction α, ∆α : W ×W ×W → W ×W ×W defined by equation of

(2.1) is a linear operator.



Int. J. Anal. Appl. 16 (1) (2018) 18

(ii) For α,β > 0, ∆α
(
∆β (xmnk)

)
= ∆α+β (xmnk) and ∆

α (∆−α (xmnk)) = xmnk.

Proof: Omitted.

Proposition 2.2. For a proper fraction α and f be an Musielak-Orlicz function, if χ3f (x) is a linear space,

then χ
3∆αγ
f (x) is also a linear space.

Proof: Omitted

3. Definitions and Preliminaries

Throughout the article w3,χ3 (∆) , Λ3 (∆) denote the spaces of all, triple gai difference sequence spaces

and triple analytic difference sequence spaces respectively.

Subramanian et al. (see [12]) introduced by a triple entire sequence spaces, triple analytic sequences spaces

and triple gai sequence spaces. The triple sequence spaces of χ3 (∆) , Λ3 (∆) are defined as follows:

χ3 (∆) =
{
x ∈ w3 : ((m + n + k)! |∆xmnk|)

1/m+n+k → 0 asm,n,k →∞
}
,

Λ3 (∆) =
{
x ∈ w3 : supm,n,k |∆xmnk|

1/m+n+k
< ∞

}
.

Definition 3.1. An Orlicz function ([see [7]) is a function M : [0,∞) → [0,∞) which is continuous, non-

decreasing and convex with M (0) = 0, M (x) > 0, for x > 0 and M (x) → ∞ as x → ∞. If convexity of

Orlicz function M is replaced by M (x + y) ≤ M (x) + M (y) , then this function is called modulus function.

Lindenstrauss and Tzafriri ([8]) used the idea of Orlicz function to construct Orlicz sequence space.

A sequence g = (gmn) defined by

gmn (v) = sup{|v|u− (fmnk) (u) : u ≥ 0} ,m,n,k = 1, 2, · · ·

is called the complementary function of a Musielak-Orlicz function f. For a given Musielak-Orlicz function

f, [see [9] ] the Musielak-Orlicz sequence space tf is defined as follows

tf =
{
x ∈ w3 : If (|xmnk|)

1/m+n+k → 0 asm,n,k →∞
}
,

where If is a convex modular defined by

If (x) =
∑∞
m=1

∑∞
n=1

∑∞
k=1 fmnk (|xmnk|)

1/m+n+k
,x = (xmnk) ∈ tf.

We consider tf equipped with the Luxemburg metric

d (x,y) =
∑∞
m=1

∑∞
n=1

∑∞
k=1 fmnk

(
|xmnk|1/m+n+k

mnk

)
is an exteneded real number.

Definition 3.2. Let α be a proper fraction. A triple difference sequence spaces of ∆αγx =
(
∆αγxmnk

)
is said

to be ∆αγ strong Cesàro summable to 0̄ if



Int. J. Anal. Appl. 16 (1) (2018) 19

limuvw→∞
1

uvw

∑u
m=1

∑v
n=1

∑w
k=1

∣∣∆αγxmnk, 0̄∣∣ = 0. In this we write ∆αγxmnk →[C,1,1,1] ∆αγxmnk. The set of
all ∆αγ strong Cesàro summable triple sequence spaces is denoted by [C, 1, 1, 1].

Definition 3.3. Let α be a proper fraction and β be a nonnegative real number. A triple difference sequence

spaces of ∆αγx =
(
∆αγxmnk

)
is said to be ∆αγ rough strong Cesàro summable in probability to a random

variable ∆αγx : W ×W ×W → R×R×R with respect to the roughness of degree β if for each � > 0,

limuvw→∞
1

uvw

∑u
m=1

∑v
n=1

∑w
k=1 P

(∣∣∆αγxmnk, 0̄∣∣ ≥ β + �) = 0. In this case we write ∆αγxmnk →[C,1,1,1]P∆β
∆αγxmnk. The class of all β∆

α
γ− strong Cesàro summable triple sequence spaces of random variables in

proability and it will be denoted by β [C, 1, 1, 1]
P∆

.

4. Rough cesÀro summable of triple of ∆αγ

In this section by using the operator ∆αγ , we introduce some new triple difference sequence spaces of rough

Cesàro summable involving lacunary sequences θ and arbitrary sequence p = (prst) of strictly positive real

numbers.

If α be a proper fraction and β be nonnegative real number. A triple difference sequence spaces of

∆αγX =
(
∆αγxmnk

)
is said to be ∆αγ− rough strong Cesàro summable in probability to a random vari-

able ∆αγX : W ×W ×W → R×R×R with respect to the roughness of degree β if for each � > 0 then define

the triple difference sequence spaces as follows: (i)

C
(
∆αγ ,p

)
θ

=
∑∞
r=1

∑∞
s=1

∑∞
t=1 P

(
fmnk

[∣∣∣ 1hrst ∑(mnk)∈Irst ∆αγX∣∣∣prst] ≥ β + �) < ∞. In this case we write
C
(
∆αγ ,p

)
θ
→[C,1,1,1]

P∆

β C
(
∆αγ ,p

)
θ
. The class of all βC

(
∆αγ ,p

)
θ
− rough strong Cesàro summable triple

sequence spaces of random variables in probability and it will be denoted by β [C, 1, 1, 1]
P∆

.

(ii)

C
[
∆αγ ,p

]
θ

=
∑∞
r=1

∑∞
s=1

∑∞
t=1 P

(
1

hrst

∑
(mnk)∈Irst fmnk

[∣∣∆αγX∣∣prst] ≥ β + �) < ∞. In this case we write
C
[
∆αγ ,p

]
θ
→[C,1,1,1]

P∆

β C
[
∆αγ ,p

]
θ
. The class of all βC

[
∆αγ ,p

]
θ
− rough strong Cesàro summable triple se-

quence spaces of random variables in probability.

(iii)

CΛ
(
∆αγ ,p

)
θ

= P
(
fmnk

[∣∣∣ 1hrst ∑(mnk)∈Irst ∆αγX∣∣∣prst] ≥ β + �) < ∞. In this case we write CΛ (∆αγ ,p)θ →[C,1,1,1]P∆β
CΛ
(
∆αγ ,p

)
θ
. The class of all βCΛ

(
∆αγ ,p

)
θ
− rough strong Cesàro summable triple sequence spaces of ran-

dom variables in probability.

(iv)

CΛ
[
∆αγ ,p

]
θ

= 1
hrst

∑
(mnk)∈Irst P

(
fmnk

[∣∣∆αγX∣∣prst] ≥ β + �) < ∞. In this case we write CΛ [∆αγ ,p]θ →[C,1,1,1]P∆β
CΛ
[
∆αγ ,p

]
θ
. The class of all βCΛ

[
∆αγ ,p

]
θ
− rough strong Cesàro summable triple sequence spaces of random

variables in probability.

(v)

N
(
∆αγ ,p

)
θ

= limrst→∞
1

hrst

∑
(mnk)∈Irst P

(
fmnk

[∣∣∆αγX, 0̄∣∣prst] ≥ β + �) = 0. In this case we write N (∆αγ ,p)θ →[C,1,1,1]P∆β



Int. J. Anal. Appl. 16 (1) (2018) 20

N
(
∆αγ ,p

)
θ
. The class of all βN

(
∆αγ ,p

)
θ
− rough strong Cesàro summable triple sequence spaces of random

variables in probability.

Theorem 4.1. If α be a proper fraction, β be nonnegative real number, f be an Musielak-Orlicz function and

(prst) is a triple difference analytic sequence then the sequence spaces C
(
∆αγ ,p

)
θ
, C

[
∆αγ ,p

]
θ
, CΛ

(
∆αγ ,p

)
θ
,

CΛ
[
∆αγ ,p

]
θ

and N
(
∆αγ ,p

)
θ

are linear spaces.

Proof: Because the linearity may be proved in a similar way for each of the sets of triple sequences, hence

it is omitted.

Theorem 4.2. If α be a proper fraction, β be nonnegative real number, f be an Musielak-Orlicz function

and (prst) , for all r,s,t ∈ N, then the triple difference sequence spaces C
[
∆αγ ,p

]
θ

is a BK-space with the

luxemburg metric is defined by

d (x,y)1 =
∑∞
u=0

∑∞
v=0

∑∞
w=0 fmnk

[
γuvwxuvw

uvw

]
+

limuvw→∞
1

uvw

∑∞
r=1

∑∞
s=1

∑∞
t=1 fmnk

[
P
(

1
hrst

∑
(m,n,k)∈Irst

∣∣∆αγx∣∣p) ≥ β + �]1/p , 1 ≤ p.
Also if prst = 1 for all (r,s,t) ∈ N, then the triple difference spaces CΛ

[
∆αγ ,p

]
θ

and N
(
∆αγ ,p

)
θ

are BK-

spaces with the luxemburg metric is defined by

d (x,y)2 =
∑∞
u=0

∑∞
v=0

∑∞
w=0 fmnk

[
γuvwxuvw

uvw

]
+

limuvw→∞
1

uvw
1

hrst

∑
(m,n,k)∈Irst fmnk

[
P
(∣∣∆αγx∣∣) ≥ β + �] .

Proof: We give the proof for the space CΛ
[
∆αγ ,p

]
θ

and that of others followed by using similar techniques.

Suppose (xn) is a Cauchy sequence in CΛ
[
∆αγ ,p

]
θ
, where xn = (xij`)

n
and xm =

(
xmij`

)
are two elements

in CΛ
[
∆αγ ,p

]
θ
. Then there exists a positive integer n0 (�) such that |xn −xm| → 0 as m,n → ∞. for all

m,n ≥ n0 (�) and for each i,j,` ∈ N. Therefore


x11uvw x
12
uvw ... ...

x21uvw x
22
uvw ... ...

.

.

.




and




∆αγx
11
ij` ∆

α
γx

12
ij` ... ...

∆αγx
21
ij` ∆

α
γx

22
ij` ... ...

.

.

.




are Cauchy sequences in complex field C and CΛ
[
∆αγ ,p

]
θ

respectively. By using the completness of C and

CΛ
[
∆αγ ,p

]
θ

we have that they are convergent and suppose that xnij` → xij` in C and
(

∆αγx
n
ij`

)
→ yij` in

CΛ
[
∆αγ ,p

]
θ

for each i,j,` ∈ N as n → ∞. Then we can find a triple sequence space of (xij`) such that

yij` = ∆
α
γxij` for i,j,` ∈ N. These xsij` can be interpreted as

xij` =
1
γij`

∑i−m
u=1

∑j−n
v=1

∑`−k
w=1 ∆

α
γyuvw =

1
γij`

∑i
u=1

∑j
v=1

∑`
w=1 ∆

α
γyu−m,v−n,w−k, (y1−m,1−n,1−k = y2−m,2−n,2−k = · · · = y000 = 0) . for sufficiently large

(i,j,`) ; that is,



Int. J. Anal. Appl. 16 (1) (2018) 21

(
∆αγx

n
)

=




∆αγx
11
ij` ∆

α
γx

12
ij` ... ...

∆αγx
21
ij` ∆

α
γx

22
ij` ... ...

.

.

.




converges to
(
∆αγxij`

)
for each i,j,` ∈ N as n →∞. Thus |xm −x|2 → 0 as m →∞. Since CΛ

[
∆αγ ,p

]
θ

is

a Banach luxemburg metric with continuous coordinates, that is |xn −x|2 → 0 implies
∣∣∣xnij` −xij`∣∣∣ → 0 for

each i,j,` ∈ N as n →∞, this shows that CΛ
[
∆αγ ,p

]
θ

is a BK-space.

Theorem 4.3. If α be a proper fraction, β be nonnegative real number, f be an Musielak-Orlicz function

and (prst) , for all r,s,t ∈ N, then the triple difference sequence space C
(
∆αγ ,p

)
θ

is a BK-space with the

luxemburg metric is defined by

d (x,y)3 =
∑∞
u=0

∑∞
v=0

∑∞
w=0 fmnk

[
γuvwxuvw

uvw

]
+

limuvw→∞
1

uvw

∑∞
r=1

∑∞
s=1

∑∞
t=1 fmnk

[
P
(∣∣∣ 1hrst ∑(m,n,k)∈Irst ∆αγx∣∣∣p) ≥ β + �]1/p , 1 ≤ p.

Also if prst = 1 for all (r,s,t) ∈ N, then the triple difference spaces CΛ
(
∆αγ ,p

)
θ

is a BK-spaces with the

luxemburg metric is defined by

d (x,y)4 =
∑∞
u=0

∑∞
v=0

∑∞
w=0 fmnk

[
γuvwXuvw

uvw

]
+

limuvw→∞
1

uvw
fmnk

[
P
(∣∣∣ 1hrst ∑(m,n,k)∈Irst ∆αγx∣∣∣) ≥ β + �] .

Proof: The proof follows from Theorem 4.2.

Now, we can present the following theorem, determining some inclusion relations with out proof, since it is

a routine verification.

Theorem 4.4. Let α,ξ be two positive proper fractions α > ξ > 0 and β be two nonnegative real number,

f be an Musielak-Orlicz function and (prst) = p , for each r,s,t ∈ N be given.Then the following inclusions

are satisfied:

(i) C
(
∆ξγ,p

)
θ
⊂ C

(
∆αγ ,p

)
θ

,

(ii) C
[
∆ξγ,p

]
θ
⊂ C

[
∆αγ ,p

]
θ

,

(i) C
(
∆αγ ,p

)
θ
⊂ C

(
∆αγ ,q

)
θ
, 0 < p < q.

5. Rough Lacunary statistical convergence of triple of ∆αγ

In this section by using the operator ∆αγ , we introduce some new triple difference sequence spaces involving

rough lacunary statistical sequences spaces and arbitrary sequence p = (prst) of strictly positive real numbers.



Int. J. Anal. Appl. 16 (1) (2018) 22

Definition 5.1. The triple sequence θi,`,j = {(mi,n`,kj)} is called triple lacunary if there exist three in-

creasing sequences of integers such that

m0 = 0,hi = mi −mr−1 →∞ as i →∞ and

n0 = 0,h` = n` −n`−1 →∞ as ` →∞.

k0 = 0,hj = kj −kj−1 →∞ as j →∞.

Let mi,`,j = min`kj,hi,`,j = hih`hj, and θi,`,j is determine by

Ii,`,j = {(m,n,k) : mi−1 < m < mi andn`−1 < n ≤ n` andkj−1 < k ≤ kj} ,qi = mimi−1 ,q` =
n`
n`−1

,qj =
kj
kj−1

.

Definition 5.2. Let α be a proper fraction, f be an Musielak-Orlicz function and θ = {mrnskt}(rst)∈N ⋃ 0
be the triple difference lacunary sequence spaces of

(
∆αγXmnk

)
is said to be ∆αγ− lacunary statistically

convergent to a number 0̄ if for any � > 0,

limrst→∞
1

hrst

∣∣{(m,n,k) ∈ Irst : fmnk [∣∣∆αγXmnk, 0̄∣∣] ≥ �}∣∣ = 0 , where
Ir,s,t = {(m,n,k) : mr−1 < m < mr andns−1 < n ≤ ns andkt−1 < k ≤ kt} ,qr = mrmr−1 ,qs =

ns
ns−1

,qt =
kt
kt−1

.

In this case write ∆αγX →Sθ ∆αγx.

Definition 5.3. If α be a proper fraction, β be nonnegative real number,f be an Musielak-Orlicz function

and θ = {mrnskt}(rst)∈N ⋃ 0 be the triple difference sequence spaces of lacunary. A number X is said to be
∆αγ −Nθ− convergent to a real number 0̄ if for every � > 0,

limrst→∞
1

hrst

∑
m∈Ir

∑
n∈Is

∑
k∈It fmnk

[∣∣∆αγXmnk, 0̄∣∣] = 0. In this case we write ∆αγXmnk →Nθ 0̄.
Definition 5.4. Let α be a proper fraction, β be nonnegative real number,f be an Musielak-Orlicz function

and arbitary sequence p = (prst) of strictly positive real numbers. A triple difference sequence spaces of

random variables is said to be ∆αγ− rough lacunary statistically convergent in probability to ∆αγX : W ×W ×

W → R×R×R with respect to the roughness of degree β if for any �,δ > 0, limrst→∞
1

hrst

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �) ≥ δ}∣∣ = 0 and we write ∆αγXmnk →SPβ 0̄. It
will be denoted by βSPθ .

Definition 5.5. Let α be a proper fraction, β be nonnegative real number,f be an Musielak-Orlicz function

and arbitary sequence p = (prst) of strictly positive real numbers. A triple difference sequence spaces of

random variables is said to be ∆αγ− rough Nθ− convergent in probability to ∆αγX : W ×W ×W → R×R×R

with respect to the roughness of degree β if for any � > 0, limrst→∞
1

hrst

∑
m∈Ir

∑
n∈Is

∑
k∈It∣∣{P ([fmnk (∣∣∆αγXmnk∣∣)]prst ≥ β + �)}∣∣ = 0, and we write ∆αγXmnk →NPθβ ∆αγX. The class of all β −Nθ−

convergent triple difference sequence spaces of random variables in probability will be denoted by βNPθ .

Definition 5.6. Let α be a proper fraction, β be nonnegative real number,f be an Musielak-Orlicz function

and arbitary sequence p = (prst) of strictly positive real numbers. A triple difference sequence spaces of

random variables is said to be ∆αγ− rough lacunary statistically Cauchy if there exists a number N = N (�)



Int. J. Anal. Appl. 16 (1) (2018) 23

in probability to ∆αγX : W ×W ×W → R×R×R with respect to the roughness of degree β if for any �,δ > 0,

limrst→∞

1
hrst

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγ (xmnk −xN )∣∣)]prst ≥ β + �) ≥ δ}∣∣ = 0.
Theorem 5.1. Let α be a proper fraction, β be nonnegative real number,f be an Musielak-Orlicz function

and arbitary sequence p = (prst) of strictly positive real numbers, 0 < p < ∞. (i) If (xmnk) →
(
N
(
∆αγ ,p

)
θ

)
for prst = p then (xmnk) →

(
∆αγ (Sθ)

)
. (ii) If x ∈

(
∆αγ (Sθ)

)
, then (xmnk) →

(
N
(
∆αγ ,p

)
θ

)
.

Proof: Let x = (xmnk) ∈
(
N
(
∆αγ ,p

)
θ

)
and � > 0,

∣∣{P ([fmnk (∣∣∆αγXmnk∣∣)]prst ≥ β + �)}∣∣ = 0. We have
1

hrst

∑
(mnk)∈Irst

∣∣{P ([fmnk (∣∣∆αγXmnk∣∣)]prst ≥ β + �)}∣∣ ≥
1

hrst

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �) ≥ δ}∣∣(β+�δ )p .
So we observe by passing to limit as r,s,t →∞,

limrst→∞
1

hrst

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �) ≥ δ}∣∣ ≤(
δ

α+�

)p
P
(
limrst→∞

1
hrst

∑
(m,n,k)∈Irst

∣∣∆αγxmnk∣∣p) = 0. which implies that xmnk → (∆αγ (Sθ)) .
Suppose that x ∈ ∆αγ

(
Λ3
)

and (xmnk) →
(
∆αγ (S)

)
. Then it is obvious that

(
∆αγx

)
∈ Λ3 and

1
hrst

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �) ≥ δ}∣∣ → 0 as r,s,t → ∞. Let � > 0 be given
and there exists u0v0w0 ∈ N such that∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �2) ≥ δ2}∣∣ ≤ �2(d(∆αγx,y))Λ3 + δ2,
where

∑∞
u=1

∑∞
v=1

∑∞
w=1 |γuvwxuvw| = 0, for all r ≥ u0,s ≥ v0, t ≥ w0. Further more, we can write∣∣∆αγxmnk∣∣ ≤ d(∆αγxmnk,y)∆αγ ≤ d(∆αγx,y)Λ3 = d (x,y)∆αγx. For r,s,t ≥ u0,v0,w0.

1
hrst

∑
(mnk)∈Irst P

([
fmnk

(∣∣∆αγXmnk∣∣)]p) = 1hrst P (∑(mnk)∈Irst [fmnk (∣∣∆αγXmnk∣∣)]p) +
1

hrst
P
(∑

(mnk)/∈Irst

[
fmnk

(∣∣∆αγXmnk∣∣)]p) < 1hrst P
(
hrst

(
�
2

+ δ
2

)
+ hrst

� d(x,y)
p
∆αγX

2 d(x,y)
p
∆αγX

+ δ
2

)
= � + δ. Hence

(xmnk) →
(
N
(
∆αγ ,p

)
θ

)
.

Corollary 5.1. If α be a proper fraction, β be nonnegative real number,f be an Musielak-Orlicz function

and arbitary sequence p = (prst) of strictly positive real numbers then the following statements are hold:

(i) S
⋂

Λ3 ⊂ ∆αγ (Sθ)
⋂

∆αγ
(
Λ3
)
, (ii) ∆αγ (Sθ)

⋂
∆αγ
(
Λ3
)

= ∆αγ
(
w3p
)
.

Theorem 5.2. Let α be a proper fraction, β be nonnegative real number,f be an Musielak-Orlicz function

and arbitary sequence p = (prst) of strictly positive real numbers. if x = (xmnk) is a ∆
α
γ− triple difference

rough lacunary statistically convergent sequence, then x is a ∆αγ− triple difference rough lacunary statistically

Cauchy sequence.

Proof: Assume that (xmnk) →
(
∆αγ (Sθ)

)
and �,δ > 0. Then

1
δ

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγxmnk∣∣)]prst ≥ β + �2)}∣∣ for almost all m,n,k and if we select N, then
1
δ

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγxN∣∣)]prst ≥ β + �2)}∣∣ holds. Now, we have∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγ (xmnk −xN )∣∣)]prst)}∣∣ ≤
1
δ

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγxmnk∣∣)]prst ≥ β + �2)}∣∣ +



Int. J. Anal. Appl. 16 (1) (2018) 24

1
δ

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγxN∣∣)]prst ≥ β + �2)}∣∣ < 1δ (β + �) = �, for almost m,n,k. Hence (xmnk)
is a ∆αγ− rough lacunary statistically Cauchy.

Theorem 5.3. If α be a proper fraction, β be nonnegative real number,f be an Musielak-Orlicz function and

arbitary sequence p = (prst) of strictly positive real numbers and 0 < p < ∞, then N
(
∆αγ ,p

)
θ
⊂ ∆αγ (Sθ) .

Proof: Suppose that x = (xmnk) ∈ N
(
∆αγ ,p

)
θ

and∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγxmnk∣∣)]p ≥ β + �)}∣∣. Therefore we have
1

hrst

∑
(mnk)∈Irst P

([
fmnk

(∣∣∆αγxmnk∣∣)]p) ≥ 1hrst ∑(mnk)∈Irst (β + �)p ≥
1

hrst

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγxmnk∣∣)]p ≥ β + �)}∣∣ (β + �)p .
So we observe by passing to limit as r,s,t →∞,

limrst→∞
1

hrst

∣∣{(m,n,k) ∈ Irst : P ([fmnk (∣∣∆αγ (xmnk)∣∣)]p ≥ β + �) ≥ δ}∣∣ <
1

(β+�)p

(
P
(
limrst→∞

1
hrst

∑
(m,n,k)∈Irst

[
fmnk

(∣∣∆αγ (xmnk)∣∣)]p)) = 0 implies that
x ∈ ∆αγ (Sθ) . Hence N

(
∆αγ ,p

)
θ
⊂ ∆αγ (Sθ) .

Competing Interests: The authors declare that there is not any conflict of interests regarding the

publication of this manuscript.

References

[1] A. Esi , On some triple almost lacunary sequence spaces defined by Orlicz functions, Res. Rev., Discr. Math. Struct., 1(2)

(2014), 16-25.

[2] A. Esi and M. Necdet Catalbas,Almost convergence of triple sequences, Glob. J. Math. Anal., 2(1) (2014), 6-10.

[3] A. Esi and E. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space,Appl. Math. Inf.

Sci., 9 (5) (2015), 2529-2534.

[4] A. J. Datta A. Esi and B.C. Tripathy,Statistically convergent triple sequence spaces defined by Orlicz function , J. Math.

Anal., 4(2) (2013), 16-22.

[5] S. Debnath, B. Sarma and B.C. Das ,Some generalized triple sequence spaces of real numbers , J. Nonlinear Anal. Optim.,

6(1) (2015), 71-79.

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

[7] P.K. Kamthan and M. Gupta, Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel

Dekker, Inc. New York , 1981.

[8] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390.

[9] J. Musielak, Orlicz Spaces,Lectures Notes in Math.,1034, Springer-Verlag, 1983.

[10] A. Sahiner, M. Gurdal and F.K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math. , 8 No.

(2)(2007), 49-55.

[11] A. Sahiner, B.C. Tripathy , Some I related properties of triple sequences, Selcuk J. Appl. Math. , 9(2)(2008), 9-18.

[12] N. Subramanian and A. Esi, The generalized tripled difference of χ3 sequence spaces, Glob. J. Math. Anal., 3(2) (2015), 54-

60.


	1. Introduction
	2. Some new difference triple sequence spaces with fractional order
	3. Definitions and Preliminaries
	4. Rough cesro summable of triple of 
	5. Rough Lacunary statistical convergence of triple of 
	References