Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
DIFFERENCE
SEQUENCES
OF
BI-COMPLEX
UNCERTAIN
SEQUENCES
∗
Donka
Kalawathi
†
Binod
Chandra
Tripathy
‡
DOI 10.56082/annalsarscimath.2026.4.19
Abstract
In
this
paper,
we
apply
the
concept
of
difference
operators
to
se-
quence
spaces
in
the
framework
of
uncertainty
theory.
Using
these
concepts,
we
introduce
new
classes
of
convergent
difference
sequences
spaces
involving
bi-complex
uncertain
variables,
and
we
examine
their
algebraic
and
topological
properties,
including
linearity
and
normed
space
structures.
Furthermore,
we
establish
inclusion
relations
among
these
spaces,
particularly
strict
inclusion
∆
n
−
1
⊂
∆
n
,
and
exam-
ine
important
structural
characteristics
such
as
solidity,
monotonicity,
convergence-free,
sequence
algebra
and
symmetry,
showing
that
these
properties
generally
do
not
hold.
We
also
study
convergence
relations
and
implications
associated
with
the
newly
defined
sequence
spaces.
Several
results
and
examples
are
provided
to
support
the
developed
theory.
Keywords:
bi-complex
uncertain
variable,
convergence,
difference
se-
quence
space,
solid
space,
symmetric
space.
MSC:
40A05,
40A30,
40C15,
40F05,
60B10,
60D05,
60E15,
60F15.
1
Introduction
In
functional
analysis,
especially
in
the
study
of
sequence
spaces,
the
con-
cept
of
sequence
convergence
plays
an
important
and
significant
role.
This
∗
Accepted
for
publication
on
July
01,
2026
†
kala9285@gmail.com
,
Tripura
University,
Suryamaninagar,
Agartala,
India
‡
tripathybc@gmail.com,
tripathybc@yahoo.com
,
Tripura
University,
Suryamanina-
gar,
Agartala,
India
19
Difference
Sequences
of
Bi-complex
Uncertain
Sequences
20
idea
has
been
discussed
in
earlier
works
by
Rath
and
Tripathy
[
13
].
Un-
certainty
theory
was
introduced
by
Liu
[
10
]
in
2007
using
the
idea
of
an
uncertain
measure,
which
satisfies
properties
such
as
normality,
duality,
sub-
additivity,
and
product
axioms.
Later,
Liu
[
11
]
presented
a
revised
version
of
this
theory
in
2016.
You
[
21
]
developed
the
notion
of
uniform
almost
sure
convergence
and
examined
its
connections
with
various
other
modes
of
convergence.
Zhang
[
22
]
studied
several
important
results
concerning
the
convergence
behavior
of
uncertain
sequences
were
established.
Furthermore,
Guo
and
Xu
[
4
]
investigated
the
conditions
for
mean
square
convergence
of
uncertain
sequences.
The
idea
of
bi-complex
numbers
was
first
proposed
by
Segre
and
later
explored
by
several
researchers
such
as
Wagh
[
20
],
Srivastava
and
Srivas-
tava
[
18
],
Sager
and
Sager
[
15
],
Rochon
and
Shapiro
[
14
],
Beg
et
al.
[
1
]
and
Singh
[
17
].
However,
the
most
detailed
and
foundational
work
in
this
area
was
done
by
Price
[
12
]
and
later,
in
1991,
he
published
a
comprehensive
book
covering
bi-complex
numbers,
multi-complex
spaces,
and
their
func-
tion
theory.
Recent
studies
have
explored
special
bi-complex
sequences
and
related
number
systems.
Kizilates
et
al.
[
5
]
studied
bi-complex
general-
ized
Tribonacci
quaternions.
Further
developments
on
hybrid
and
hyper-
complex
sequences
were
reported
by
Kizilates
[
6
,
7
]
and
Kizilates
and
Kone
[
8
].
This
study
motivates
further
investigation
of
sequence
spaces
involving
bi-complex
structures.
The
concept
of
difference
sequences
was
first
introduced
by
Kizmaz.
[
9
]
After
that,
Esi
et
al.
[
2
]
developed
generalized
difference
sequence
spaces
as
follows:
Let
m,
n
≥
0
be
fixed
integers,
Z
(∆
n
m
)
=
{
y
=
(
y
k
)
∈
ω
:
∆
n
m
y
=
(∆
n
m
y
k
)
∈
Z
}
,
for
Z
=
∞
,
c
and
c
0
;
where
∆
n
m
y
k
=
(∆
n
−
1
m
y
k
−
∆
n
−
1
m
y
k
+
m
)
and
∆
0
m
y
k
=
y
k
for
all
n
∈
N
.
This
notion
of
generalized
difference
gave
the
binomial
representation
as
follows:
∆
n
m
y
k
=
n
v
=0
(
−
1)
v
n
v
y
k
+
mv
for
all
k
∈
N
.
For
m
=
1
and
n
=
1
,
we
have
the
spaces
∞
(∆)
,
c
(∆)
and
c
0
(∆)
introduced
and
studied
by
Kizmaz.
[
9
].
For
m
=
1
,
we
have
the
spaces
D.
Kalawathi,
B.C.
Tripathy
21
∞
(∆
n
)
,
c
(∆
n
)
and
c
0
(∆
n
)
introduced
and
studied
by
Et
and
Colak
[
3
].
For
n
=
1
,
we
have
the
spaces
∞
(∆
m
)
,
c
(∆
m
)
and
c
0
(∆
m
)
introduced
and
studied
by
Tripathy
and
Esi
[
19
].
The
sequence
spaces
Z
(∆
n
m
)
for
Z
=
∞
,
c
and
c
0
are
Banach
spaces,
by
the
norm
y
∆
n
m
=
p
i
=1
|
y
i
|
+
sup
k
|
∆
n
m
y
k
|
,
where
p
=
mn
for
m
≥
1
,
n
≥
1
.
2
Definitions
and
preliminaries
We
review
basic
definitions
and
prior
results
before
the
main
contribution.
These
are
essential
for
the
study.
Definition
1.
[
10
]
Suppose,
A
is
a
σ
-algebra
on
a
nonempty
set
S
.
A
set-
valued
function
N
is
called
an
uncertain
measure
if
it
satisfies
the
following
four
axioms:
1.
Normality
Axiom:
N
(
S
)
=
1,
2.
Duality
axiom:
For
each
event
H
∈
A
,
N
(
H
)
+
N
(
H
c
)
=
1,
3.
Subadditivity
Axiom:
If
{
d
m
}
is
a
countable
collection
of
events
in
A
,
then
N
∞
m
=1
d
m
≤
∞
m
=1
N
(
d
m
)
.
4.
Product
Axiom:
Suppose
(
S
i
,
A
i
,
N
i
)
is
the
uncertainty
space
for
i
=
1,2,3,...
.
So
the
product
uncertain
measure
N
fulfill
N
∞
i
=1
d
i
=
∞
i
=1
N
i
(
d
i
)
where
each
d
i
∈
A
i
.
A
triplet
(
S
,
A
,
N
)
is
known
as
an
uncertainty
space,
and
every
H
in
A
is
known
as
an
event.
Definition
2.
[
10
]
An
uncertain
variable
is
a
measurable
function
ϑ
:
(
S
,
A
,
N
)
→
R
such
that
for
any
Borel
set
B
⊂
R
,
the
set
{
ϑ
∈
B
}
=
{
ω
∈
S
:
ϑ
(
ω
)
∈
B
}
is
an
event
in
A
.
Difference
Sequences
of
Bi-complex
Uncertain
Sequences
22
Definition
3.
A
sequence
space
X
is
said
to
be
solid
(or
normal)
if
(
α
n
u
n
)
∈
X
,
whenever
(
u
n
)
∈
X
,
for
all
sequences
(
α
n
)
of
scalars
such
that
|
α
n
|
≤
1
for
all
n
∈
N
.
Definition
4.
A
sequence
space
X
is
said
to
be
monotone
if
X
contains
the
canonical
pre-images
of
all
its
step
spaces.
Definition
5.
A
sequence
space
X
is
said
to
be
convergence
free
if
(
v
n
)
∈
X
whenever
(
u
n
)
∈
X
and
v
n
=
0
whenever
u
n
=
0
.
Definition
6.
A
sequence
space
X
is
said
to
be
a
sequence
algebra
if
(
u
n
v
n
)
∈
X
whenever
(
u
n
)
∈
X
and
(
v
n
)
∈
X
.
Definition
7.
A
sequence
space
X
is
said
to
be
symmetric
if
(
u
k
)
∈
X
→
(
u
π
(
k
))
∈
X,
where
π
is
a
permutation
of
N
.
3
Bi-complex
difference
sequence
Let
(
ϑ
r
)
BC
be
a
sequence
of
bi-complex
uncertain
variables
in
(
S
,
A
,
N
)
where
ϑ
r
=
ξ
r
+
i
1
η
r
+
i
2
µ
r
+
i
1
i
2
ϕ
r
.
(
U,
∆
n
m
)
=
{
ϑ
=
(
ϑ
r
(
ω
))
BC
:
there
exists
ϑ
(
ω
)
such
that
1
p
p
r
=1
∆
n
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
→
0
,
as
p
→
∞}
.
Using
the
difference
operator,
we
have
defined
some
new
classes
of
sequences
of
bi-complex
uncertain
variables
as
follows:
c
0
(
U,
∆
n
m
)
BC
=
ϑ
:
lim
r
→∞
∆
n
m
ϑ
r
(
ω
)
BC
=
0
.
c
(
U,
∆
n
m
)
BC
=
ϑ
:
lim
r
→∞
∆
n
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
=
0
for
some
bi-complex
uncertain
variable
ϑ
(
ω
)
.
∞
(
U,
∆
n
m
)
BC
=
ϑ
:
sup
r
∆
n
m
ϑ
r
(
ω
)
BC
<
∞
.
(
U,
∆
n
m
)
BC
=
ϑ
:
lim
p
→∞
1
p
p
j
=1
∆
n
m
ϑ
k
+
j
(
ω
)
−
ϑ
(
ω
)
BC
=
0
,
uniformly
in
k
.
D.
Kalawathi,
B.C.
Tripathy
23
Theorem
1.
The
sets
c
0
(
U,
∆
n
m
)
BC
,
c
(
U,
∆
n
m
)
BC
and
∞
(
U,
∆
n
m
)
BC
are
lin-
ear
spaces.
Proof.
We
examine
the
result
for
the
class
of
bi-complex
uncertain
sequences
c
0
(
U,
∆
n
m
)
BC
.
The
other
cases
can
be
treated
in
a
similar
way.
Let
(
ϑ
r
)
,
(
ϑ
r
)
∈
c
0
(
U,
∆
n
m
)
BC
.
Then,
lim
r
→∞
∆
n
m
ϑ
r
(
ω
)
BC
=
0
lim
r
→∞
∆
n
m
ϑ
r
(
ω
)
BC
=
0
Now,
for
α,
β
∈
BC
,
lim
r
→∞
∆
n
m
(
αϑ
r
(
ω
)
+
βϑ
r
(
ω
))
BC
=
lim
r
→∞
α
∆
n
m
ϑ
r
(
ω
)
+
β
∆
n
m
ϑ
r
(
ω
)
BC
≤
|
α
|
lim
r
→∞
∆
n
m
ϑ
r
(
ω
)
BC
+
|
β
|
lim
r
→∞
∆
n
m
ϑ
r
(
ω
)
BC
,
as
r
→
∞
.
Hence,
(
αϑ
r
(
ω
)
+
βϑ
r
(
ω
))
∈
c
0
(
U,
∆
n
m
)
BC
.
Consequently,
c
0
(
U,
∆
n
m
)
BC
forms
a
linear
space.
The
same
argument
proves
that
c
(
U,
∆
n
m
)
BC
and
∞
(
U,
∆
n
m
)
BC
are
also
linear
spaces.
Theorem
2.
The
sets
c
0
(
U,
∆
n
m
)
BC
,
c
(
U,
∆
n
m
)
BC
and
∞
(
U,
∆
n
m
)
BC
are
normed
linear
spaces,
normed
by
ϑ
r
(
ω
)
(∆
n
m
)
BC
=
j
q
=1
ϑ
q
(
ω
)
+
sup
r
∆
n
m
ϑ
r
(
ω
)
where
j
=
mn
for
m
≥
1
,
n
≥
1
.
Proof.
Let
ϑ
r
∈
∞
(
U,
∆
n
m
)
BC
and
suppose
that
ϑ
(
ω
)
(∆
n
m
)
BC
=
0
.
Then
j
q
=1
ϑ
q
(
ω
)
+
sup
r
∆
n
m
ϑ
r
(
ω
)
=
0
.
Difference
Sequences
of
Bi-complex
Uncertain
Sequences
24
Since
every
term
is
non-negative,
it
follows
that
ϑ
q
(
ω
)
=
0
,
for
q
=
1
,
2
,
...,
mn
and
∆
n
m
ϑ
r
(
ω
)
=
0
.
For
r
=
1
,
∆
n
m
ϑ
1
(
ω
)
=
0
which
implies
∆
n
−
1
m
ϑ
1
(
ω
)
−
∆
n
−
1
m
ϑ
m
+1
(
ω
)
=
0
.
Since,
ϑ
q
(
ω
)
=
0
,
for
q
=
1
,
2
,
...,
mn,
we
obtain
ϑ
m
+1
=
0.
Proceeding
inductively
in
the
usual
Kizmaz-type
man-
ner,
we
conclude
that
ϑ
r
(
ω
)
=
0
,
r
∈
N
.
Hence,
ϑ
(
ω
)
(∆
n
m
)
BC
=
0
=
⇒
ϑ
(
ω
)
=
0
.
Now,
ϑ
(
ω
)
+
ϑ
(
ω
)
(∆
n
m
)
BC
=
j
q
=1
ϑ
q
(
ω
)
+
ϑ
q
(
ω
)
+
sup
r
∆
n
m
(
ϑ
r
(
ω
)
+
ϑ
r
(
ω
))
≤
j
q
=1
ϑ
q
(
ω
)
+
sup
r
∆
n
m
ϑ
r
(
ω
)
+
j
q
=1
ϑ
q
(
ω
)
+
sup
r
∆
n
m
ϑ
r
(
ω
)
=
ϑ
(
ω
)
(∆
n
m
)
BC
+
ϑ
(
ω
)
(∆
n
m
)
BC
Finally,
we
have
for
any
λ
◦
∈
BC
,
λ
◦
ϑ
(
ω
)
(∆
n
m
)
BC
=
j
q
=1
λ
◦
ϑ
q
(
ω
)
+
sup
r
∆
n
m
(
λ
◦
ϑ
r
(
ω
))
=
|
λ
◦
|
j
q
=1
ϑ
q
(
ω
)
+
sup
r
∆
n
m
ϑ
r
(
ω
)
=
|
λ
◦
|
ϑ
(
ω
)
(∆
n
m
)
BC
.
Hence,
.
(∆
n
m
)
BC
is
a
norm
on
sets
c
0
(
U,
∆
n
m
)
BC
,
c
(
U,
∆
n
m
)
BC
,
∞
(
U,
∆
n
m
)
BC
.
This
establishes
the
theorem.
D.
Kalawathi,
B.C.
Tripathy
25
Theorem
3.
Let
m
≥
1
and
n
≥
1
.
Then
the
bi-complex
sequence
spaces
satisfy
Z
(
U,
∆
p
m
)
BC
⊂
Z
(
U,
∆
n
m
)
BC
,
for
Z
=
c,
c
0
,
∞
.
More
generally,
for
p
=
0
,
1
,
...,
n
−
1
,
we
have
Z
(
U,
∆
p
m
)
BC
⊂
Z
(
U,
∆
n
m
)
BC
for
Z
=
c,
c
0
,
∞
.
Proof.
Let
ϑ
r
∈
Z
(
U,
∆
n
−
1
m
)
BC
,
where
Z
=
c,
c
0
,
∞
.
We
show
that
ϑ
r
∈
Z
(
U,
∆
n
m
)
BC
.
For
Z
=
c
,
let
ϑ
r
∈
c
(
U,
∆
n
−
1
m
)
BC
we
have,
lim
r
→∞
∆
n
−
1
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
=
0
.
Now,
using
the
relation
∆
n
m
ϑ
r
(
ω
)
=
∆
n
−
1
m
ϑ
r
(
ω
)
−
∆
n
−
1
m
ϑ
r
+
m
(
ω
)
,
we
have
lim
r
→∞
∆
n
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
=
lim
r
→∞
∆
n
−
1
m
ϑ
r
(
ω
)
−
∆
n
−
1
m
ϑ
r
+
m
(
ω
)
−
ϑ
(
ω
)
BC
≤
lim
r
→∞
∆
n
−
1
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
+
lim
r
→∞
∆
n
−
1
m
ϑ
r
+
m
(
ω
)
−
ϑ
(
ω
)
BC
Both
limits
are
zero.
Hence,
lim
r
→∞
∆
n
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
=
0
,
which
implies
(
ϑ
r
)
∈
c
(
U,
∆
n
m
)
BC
.
Similarly
we
can
apply
the
same
argu-
ment
for
Z
=
c
0
,
∞
,
giving
Z
(
U,
∆
p
m
)
BC
⊂
Z
(
U,
∆
n
m
)
BC
.
Using
principle
of
induction,
we
obtain
Z
(
U,
∆
p
m
)
BC
⊂
Z
(
U,
∆
n
m
)
BC
for
p
=
0
,
1
,
...,
n
−
1.
Result
1.
The
bi-complex
sequence
spaces
Z
(
U,
∆
n
m
)
BC
,
for
Z
=
c,
c
0
,
∞
are
neither
solid
and
nor
monotone.
The
above
result
follows
from
the
following
examples.
Example
1.
Let
the
bi-complex
uncertainty
space
(
S
,
A
,
N
)
be
{
ω
1
,
ω
2
,
ω
3
,
ω
4
,
...
}
equipped
with
the
power
set
and
let
H
∈
A
be
an
arbitrary
event
such
that
N
(
H
)
=
sup
ω
r
∈H
{
1
r
2
+1
}
,
if
sup
ω
r
∈H
{
1
r
2
+1
}
<
1
2
,
1
−
sup
ω
r
∈H
c
{
1
r
2
+1
}
,
if
sup
ω
r
∈H
c
{
1
r
2
+1
}
<
1
2
,
1
2
,
otherwise.
Difference
Sequences
of
Bi-complex
Uncertain
Sequences
26
Consider
the
bi-complex
uncertain
variables
defined
by
(
ϑ
r
(
ω
))
BC
=
i
1
+
i
2
+
(
r
2
+
1)
i
1
i
2
,
if
ω
=
ω
r
,
0
,
otherwise,
for
r
=
1
,
2
,
...
and
ϑ
≡
θ,
where
θ
is
a
zero
element.
Then
ϑ
r
∈
c
(
U,
∆
n
m
)
BC
⊂
∞
(
U,
∆
n
m
)
BC
,
but
ϑ
r
N
r
(
H

/
∈
∞
(
U,
∆
n
m
)
BC
.
Hence,
the
spaces
Z
(
U,
∆
n
m
)
BC
,
for
Z
=
c
,
∞
are
not
solid.
Example
2.
Now,
for
proving
the
case
for
not
monotone
we
take
the
class
of
sequence
c
0
(
U,
∆
n
m
)
BC
,
we
consider
the
sequence
with
N
(
H
)
=
ω
r
∈
ω
1
4
r
.
and
consider
the
bi-complex
uncertain
variables
defined
as
(
ϑ
r
(
ω
))
BC
=
(
i
1
+
i
2
)4
r
,
if
ω
=
ω
r
,
0
,
otherwise,
Then
ϑ
r
∈
c
0
(
U,
∆
n
m
)
BC
,
but
(
ϑ
r
N
r
(
H
))
/
∈
c
0
(
U,
∆
n
m
)
BC
.
Hence
c
0
(
U,
∆
n
m
)
BC
is
not
monotone.
Consider
the
sequence
ϑ
r
in
its
pre-image
space
defined
by
(
ϑ
r
(
ω
))
BC
=
ϑ
r
,
if
ω
=
ω
r
,
0
,
otherwise,
Then
ϑ
r
/
∈
c
0
(
U,
∆
n
m
)
BC
.
Hence,
the
space
is
not
monotone.
Result
2.
The
difference
bi-complex
uncertain
sequence
space
Z
(
U,
∆
n
m
)
BC
,
for
Z
=
c,
c
0
,
∞
are
neither
convergence
free
nor
sequence
algebra.
The
above
result
follows
from
the
following
examples.
Example
3.
Consider
the
two
uncertain
bi-complex
sequences
(
ϑ
r
)
and
(
ϑ
r
)
defined
by
(
ϑ
r
(
ω
))
BC
=
1
√
r
(1
+
i
1
+
i
2
+
i
1
i
2
)
,
if
ω
=
ω
1
,
0
,
otherwise,
Hence,
(
ϑ
r
)
∈
c
(
U,
∆
n
m
)
BC
.
Now,
define
another
sequence
(
ϑ
r
)
by
(
ϑ
r
(
ω
))
BC
=
√
r
(1
+
i
1
+
i
2
+
i
1
i
2
)
,
if
ω
=
ω
1
,
0
,
otherwise,
Thus,
(
ϑ
)
/
∈
c
(
U,
∆
n
m
)
BC
.
This
shows
that
c
(
U,
∆
n
m
)
BC
is
not
convergence
free.
Similarly
the
other
cases
can
be
proved.
D.
Kalawathi,
B.C.
Tripathy
27
Example
4.
Next
for
sequence
algebra,
consider
the
above
two
sequences.
Their
product
is
(
ϑ
r
(
ω
))
BC
(
ϑ
r
(
ω
))
BC
=
(1
+
i
1
+
i
2
+
i
1
i
2
)
2
,
if
ω
=
ω
1
,
0
,
otherwise.
Since
(
ϑ
r
)
/
∈
Z
(
U,
∆
n
m
)
BC
,
their
product
(
ϑ
r
ϑ
r
)
/
∈
Z
(
U,
∆
n
m
)
BC
.
Hence,
Z
(
U,
∆
n
m
)
BC
for
Z
=
c,
c
0
,
∞
is
not
a
sequence
algebra.
Result
3.
The
bi-complex
sequence
spaces
Z
(
U,
∆
n
m
)
BC
,
for
Z
=
c,
c
0
,
∞
are
not
symmetric.
The
above
result
follows
from
the
following
example.
Example
5.
We
prove
the
result
for
Z
(
U,
∆
n
m
)
BC
for
Z
=
c
0
,
the
other
spaces
will
be
proved
accordingly.
Consider
a
bi-complex
uncertainty
space
(
S
,
A
,
N
)
as
ω
1
,
ω
2
,
ω
3
,
ω
4
,
...
with
the
power
set
and
for
any
event
H
∈
A
such
that
N
(
H
)
=
sup
ω
r
∈H
{
r
2
r
+1
}
,
if
sup
ω
r
∈H
{
r
2
r
+1
}
<
1
2
,
1
−
sup
ω
r
∈H
c
{
r
2
r
+1
}
,
if
sup
ω
r
∈H
c
{
r
2
r
+1
}
<
1
2
,
1
2
,
otherwise.
Define
the
bi-complex
uncertain
variables
by
(
ϑ
r
(
ω
t
))
BC
=
r,
if
t
=
r,
0
,
otherwise.
Then
(
ϑ
r
)
BC
∈
c
0
(
U,
∆
n
m
)
BC
.
Now
consider
the
rearrangement
of
the
se-
quence
(
ϑ
r
)
BC
by
a
permutation
π
of
N
,
defined
by
(
ϑ
π
(
r
)
)
BC
=
{
ϑ
1
,
ϑ
4
,
ϑ
9
,
ϑ
2
...
}
/
∈
c
0
(
U,
∆
n
m
)
BC
.
Therefore,
c
0
(
U,
∆
n
m
)
BC
is
not
symmetric.
4
Convergence
of
bi-complex
uncertain
difference
sequence
This
section
discusses
the
convergence
concepts
within
the
framework
of
bi-complex
uncertain
difference
sequences.
Difference
Sequences
of
Bi-complex
Uncertain
Sequences
28
Definition
8.
The
bi-complex
uncertain
sequence
(
ϑ
r
)
BC
is
said
to
be
con-
vergent
almost
surely
(a.s.)
to
ϑ
with
respect
to
difference
sequence
if
there
exists
an
event
H
with
N
(
H
)
=
1
such
that
lim
r
→∞
{
∆
n
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
}
=
0
for
every
ω
∈
H
.
Definition
9.
The
bi-complex
uncertain
sequence
(
ϑ
r
)
BC
is
said
to
be
con-
vergent
in
measure
to
ϑ
with
respect
to
difference
sequence
if
lim
r
→∞
N{
ω
∈
S
:
∆
n
m
ϑ
r
−
ϑ
BC
≥
ε
}
=
0
Definition
10.
Let
ϑ,
ϑ
1
,
ϑ
2
,
...
be
bi-complex
uncertain
variables
with
finite
expected
values.
Then
the
bi-complex
uncertain
sequence
(
ϑ
r
)
BC
is
said
to
converge
in
mean
to
ϑ
if
lim
r
→∞
E
[
ω
∈
S
:
∆
n
m
ϑ
r
−
ϑ
BC
]
=
0
.
Definition
11.
Let
D,
D
1
,
D
2
...
be
the
Bi-complex
uncertainty
distributions
of
Bi-complex
uncertain
variables
ϑ,
ϑ
1
,
ϑ
2
,
...
,
respectively.
We
say
that
the
bi-complex
uncertain
sequence
(
ϑ
r
)
BC
converges
in
distribution
to
ϑ
with
respect
to
the
difference
sequence
if
lim
r
→∞
∆
n
m
D
r
(
BC
)
−
D
(
BC
)
=
0
for
all
real
numbers
α,
β,
γ,
θ
at
which
D
(
BC
)
is
continuous.
Definition
12.
The
sequence
(
ϑ
r
)
BC
is
said
to
converge
uniformly
almost
surely
to
ϑ
if
there
exists
(
X
k
)
,
N
(
X
k
)
→
0
such
that
(
ϑ
r
)
BC
converges
uni-
formly
to
ϑ
in
S
−
X
k
,
for
any
fixed
k
∈
N
.
The
relations
among
the
convergence
concepts
of
bi-complex
uncertain
sequences
are
discussed
below.
Theorem
4.
If
the
bi-complex
uncertain
sequence
converges
in
mean
to
ϑ
with
respect
to
the
difference
sequence
then
converges
in
measure
to
ϑ
but
not
necessarily
converse.
Proof.
By
extending
markov
inequality
complex
to
bi-complex
for
any
given
number
ε
>
0,
we
have
N{
ω
∈
S
:
∆
n
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
≥
ε
)
≤
E
[
∆
n
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
]
ε
→
0
,
D.
Kalawathi,
B.C.
Tripathy
29
as
n
→
∞
.
Thus
(
ϑ
r
)
BC
converges
in
measure
to
ϑ
but
converse
of
the
above
theorem
does
not
satisfy
and
this
can
be
proved
by
considering
a
bi-complex
uncertainty
space
(
S
,
A
,
N
)
to
be
{
ω
1
,
ω
2
,
...
}
with
power
set
and
N
(
H
)
=
sup
ω
l
∈H
{
1
l
2
+1
}
,
if
sup
ω
l
∈H
{
1
l
2
+1
}
<
1
2
,
1
−
sup
ω
l
∈H
c
{
1
l
2
+1
}
,
if
sup
ω
l
∈H
c
{
1
l
2
+1
}
<
1
2
,
1
2
,
otherwise.
Consider
the
bi-complex
uncertain
variables
defined
by
(
ϑ
l
(
ω
))
BC
=
i
1
+
i
2
+
(
l
2
+
1)
i
1
i
2
,
if
ω
=
ω
l
,
0
,
otherwise,
for
l
=1,2,...
and
ϑ
≡
0
.
For
some
small
number,
ε
>
0
and
l
≥
1,
we
have
N

ω
∈
S
:
∆
n
m
ϑ
l
(
ω
)
−
ϑ
(
ω
)
BC
≥
ε

=
N

ω
∈
S
:
∆
n
m
ϑ
l
(
ω
)
BC
≥
ε

=
N
(
ω
l
)
=
1
l
2
+
1
→
0
,
as
l
→
∞
.
So,
the
sequence
(
ϑ
l
)
BC
converges
in
measure
to
ϑ
.
However,
for
each
l
∈
N
,
the
uncertainty
distribution
of
the
uncertain
variable
ϑ
l
−
ϑ
=
ϑ
l
is
D
l
(
u
)
=
0
,
if
u
<
0
,
1
−
1
l
2
+1
,
if
0
≤
u
<
l
2
+
1
,
1
,
if
u
≥
l
2
+
1
.
So
for
each
l
≥
1,
we
have
E
p
l
=1
∆
n
m
ϑ
l
(
ω
)
−
ϑ
(
ω
)
BC
=
l
2
+1
o
1
−
(1
−
1
l
2
+
1
)
du
=
1
,
That
is,
the
sequence
(
ϑ
r
)
BC
does
not
converge
in
mean
to
ϑ
with
respect
to
the
difference
sequence,
which
proves
the
theorem.
Remark
1.
Almost
sure
convergence
with
respect
to
difference
operator
does
not
imply
convergence
in
mean
or
in
measure.
Difference
Sequences
of
Bi-complex
Uncertain
Sequences
30
The
following
example
shows
this.
Example
6.
Consider
the
bi-complex
uncertainty
space
(
S
,
A
,
N
)
to
be
ω
1
,
ω
2
,
ω
3
,
ω
4
,
...
with
N
(
H
)
=
ω
r
∈
ω
1
2
r
,
and
the
bi-complex
uncertain
variables
are
defined
by
(
ϑ
r
(
ω
))
BC
=
(
i
1
i
2
)2
r
,
if
ω
=
ω
r
,
0
,
otherwise,
for
r
=
1
,
2
,
...
and
ϑ
≡
0
.
shows
that
(
ϑ
r
)
BC
converges
a.s.
to
ϑ
and
further
the
uncertainty
distribution
of
ϑ
r
BC
are
defined
by
D
r
(
u
)
=
0
,
if
u
<
0
,
1
−
1
2
r
,
if
0
≤
u
<
2
r
,
1
,
if
u
≥
2
r
,
for
r
=
1
,
2
,
...
,
respectively.
Therefore,
we
get
E
[
∆
n
m
ϑ
r
−
ϑ
BC
]
=
2
r
0
1
−
(1
−
1
2
r
)
du
=
1
,
imply
that
the
sequence
(
ϑ
n
)
BC
does
not
converge
in
mean
to
ϑ
.
Example
7.
Now,
for
proving
almost
sure
convergence
a.s.
does
not
imply
convergence
in
measure
we
consider
an
uncertainty
space
(
S
,
A
,
N
)
to
be
ω
1
,
ω
2
,
ω
3
,
ω
4
,
...
with
N
(
H
)
=
sup
ω
r
∈H
{
r
2
r
+1
}
,
if
sup
ω
r
∈H
{
r
2
r
+1
}
<
1
2
,
1
−
sup
ω
r
∈H
c
{
r
2
r
+1
<
1
2
}
,
if
sup
ω
r
∈H
c
{
r
2
r
+1
}
<
1
2
,
1
2
,
otherwise,
and
the
bi-complex
uncertain
variables
are
given
by
(
ϑ
r
(
ω
))
BC
=
(
i
1
i
2
)
r,
if
ω
=
ω
r
,
0
,
otherwise,
D.
Kalawathi,
B.C.
Tripathy
31
for
r
=
1
,
2
,
...
and
ϑ
≡
θ
,
where
θ
is
the
zero
element.
Then
we
get
N{
∆
n
m
ϑ
r
(
ω
)
−
ϑ
(
ω
)
BC
≥
ε
}
=
N{
ω
∈
S
:
∆
n
m
ϑ
r
(
ω
)
BC
≥
ε
}
=
N{
ω
r
}
=
r
2
r
+
1
→
1
2
,
as
r
→
∞
implies
that
the
sequence
(
ϑ
r
)
BC
does
not
converge
in
measure
to
ϑ
and
also
the
uncertainty
distribution
of
(
ϑ
r
)
BC
are
D
r
(
BC
)
=
D
r
(
α
+
i
1
β
+
i
2
γ
+
i
1
i
2
θ
)
=
0
,
if
α
<
0
,
β
<
0
,
γ
<
0
,
θ
<
+
∞
,
0
,
if
α
≥
0
,
β
≥
0
,
γ
≥
0
,
θ
<
0
,
1
−
r
2
r
+1
,
if
α
≥
0
,
β
≥
0
,
γ
≥
0
,
0
≤
θ
<
r,
1
,
if
α
≥
0
,
β
≥
0
,
γ
≥
0
,
θ
≥
r,
.
D
(
BC
)
=
0
,
if
α
<
0
,
β
<
0
,
γ
<
0
,
θ
<
+
∞
,
0
,
if
α
≥
0
,
β
≥
0
,
γ
≥
0
,
θ
<
0
,
1
,
if
α
≥
0
,
β
≥
0
,
γ
≥
0
,
θ
≥
0
.
Therefore,
showing
that
D
r
(
BC
)
does
not
converge
to
D
(
BC
)
at
α
≥
0
,
β
≥
0
,
γ
≥
0
,
θ
≥
0
.
That
is,
the
sequence
(
ϑ
r
)
BC
does
not
converge
to
ϑ
in
distribution.
So,
we
can
say
that
almost
surely
a.s.
convergence
with
respect
to
the
difference
operator
does
not
implies
neither
convergence
in
mean
nor
con-
vergence
in
measure
by
the
above
examples
6
and
7.
Theorem
5.
If
the
bi-complex
uncertain
sequence
(
ϑ
r
)
BC
converges
uni-
formly
almost
surely
(u.a.s.)
to
ϑ
with
respect
to
difference
operator,
then
(
ϑ
r
)
BC
converges
almost
surely
(a.s.)
to
ϑ
with
respect
to
difference
operator.
Proof.
We
know
that
if
(
ϑ
r
)
BC
converges
uniformly
almost
surely
(u.a.s.)
to
ϑ
,
then
lim
r
→∞
N
∞
r
=
k
{
∆
n
m
ϑ
r
−
ϑ
BC
≥
ε
}
=
0
.
Since,
N
∞
k
=1
∞
r
=
k
{
∆
n
m
ϑ
r
−
ϑ
BC
≥
ε
}
≤
inf
k
N
∞
r
=
k
{
∆
n
m
ϑ
r
−
ϑ
BC
≥
ε
}
.
Difference
Sequences
of
Bi-complex
Uncertain
Sequences
32
By
applying
the
limit
above
r
→
∞
,
we
get
N
∞
k
=1
∞
r
=
k
{
∆
n
m
ϑ
r
−
ϑ
BC
≥
ε
}
=
0
.
Hence,
(
ϑ
r
)
BC
converges
almost
surely
(
a.s.
)
to
ϑ
which
proves
the
theorem.
Theorem
6.
Let
the
bi-complex
uncertain
sequence
(
ϑ
r
)
BC
has
real
part
(
ξ
r
)
and
imaginary
parts
(
η
r
)
,
(
µ
r
)
and
(
ϕ
r
)
,
respectively,
for
all
n
∈
N
.
If
uncertain
sequences
ξ
r
→
ξ,
η
r
→
η,
µ
r
→
µ
and
ϕ
r
→
ϕ
are
convergent
in
measure
with
respect
to
difference
operator
on
sequence,
the
bi-complex
un-
certain
sequence
(
ϑ
r
)
converge
uniformly
with
respect
to
difference
operator
in
distribution
to
ϑ
=
ξ
+
i
1
η
+
i
2
µ
+
i
1
i
2
ϕ
.
Proof.
Let
BC
=
α
+
i
1
β
+
i
2
γ
+
i
1
i
2
θ
be
a
continuity
point
of
the
distribu-
tion
D
.
For
any
w
>
α,
x
>
β,
y
>
γ,
z
>
θ
,
we
have
{
ξ
r
≤
α,
η
r
≤
β,
µ
r
≤
γ,
ϕ
r
≤
θ
}
=
{
ξ
r
≤
α,
η
r
≤
β,
µ
r
≤
γ,
ϕ
r
≤
θ,
ξ
≤
w,
η
≤
x,
µ
≤
y,
ϕ
≤
z
}
∪
{
ξ
r
≤
α,
η
r
≤
β,
µ
r
≤
γ,
ϕ
r
≤
θ,
ξ
>
w,
η
>
x,
µ
>
y,
ϕ
>
z
}
∪
{
ξ
r
≤
α,
η
r
≤
β,
µ
r
≤
γ,
ϕ
r
≤
θ,
ξ
>
w,
η
≤
x,
µ
≤
y,
ϕ
≤
z
}
⊂
{
ξ
≤
w,
η
≤
x,
µ
≤
y,
ϕ
≤
z
}
∪
{
∆
n
m
ξ
r
(
ω
)
−
ξ
(
ω
)
BC
≥
w
−
α
}
∪
{
∆
n
m
η
r
(
ω
)
−
η
(
ω
)
BC
≥
x
−
β
}
∪
{
∆
n
m
µ
r
(
ω
)
−
µ
(
ω
)
BC
≥
y
−
γ
}
∪{
∆
n
m
ϕ
r
(
ω
)
−
ϕ
(
ω
)
BC
≥
z
−
θ
}
.
By
sub-additivity,
D
r
(
BC
)
=
D
r
(
α
+
i
1
β
+
i
2
γ
+
i
1
i
2
θ
)
≤
D
(
w
+
i
1
x
+
i
2
y
+
i
1
i
2
z
)
+
N
(
∆
n
m
ξ
r
(
ω
)
−
ξ
(
ω
)
BC
≥
w
−
α
)
+
N
(
∆
n
m
η
r
(
ω
)
−
η
(
ω
)
BC
≥
x
−
β
)
+
N
(
∆
n
m
µ
r
(
ω
)
−
µ
(
ω
)
BC
≥
y
−
γ
)
+
N
(
∆
n
m
ϕ
r
(
ω
)
−
ϕ
(
ω
)
BC
≥
z
−
θ
)
.
Given
each
(
ξ
r
)
→
ξ,
(
η
r
)
→
η,
(
µ
r
)
→
µ
and
(
ϕ
r
)
→
ϕ
converge
in
measure,
so
lim
r
→∞
N{
ω
∈
S
:
∆
n
m
ξ
r
(
ω
)
−
ξ
(
ω
)
BC
≥
w
−
α
=
0
}
,
D.
Kalawathi,
B.C.
Tripathy
33
lim
n
→∞
N{
ω
∈
S
:
∆
n
m
η
r
(
ω
)
−
η
(
ω
)
BC
≥
x
−
β
=
0
}
,
lim
n
→∞
N{
ω
∈
S
:
∆
n
m
µ
r
(
ω
)
−
µ
(
ω
)
BC
≥
y
−
γ
=
0
}
,
and
lim
r
→∞
N{
ω
∈
S
:
∆
n
m
ϕ
r
(
ω
)
−
ϕ
(
ω
)
BC
≥
z
−
θ
=
0
}
.
Also
lim
r
→∞
sup
D
r
(
BC
)
≤
D
(
w
+
i
1
x
+
i
2
y
+
i
1
i
2
z
)
,
for
any
w
>
α,
x
>
β,
y
>
γ
and
z
>
θ
and
w
+
i
1
x
+
i
2
y
+
i
1
i
2
z
→
α
+
i
1
β
+
i
2
γ
+
i
1
i
2
θ
,
we
obtain
lim
r
→∞
sup
D
r
(
BC
)
≤
D
(
BC
)
.
(1)
And
for
p
<
α,
λ
<
β,
κ
<
γ,
q
<
θ
,
{
ξ
≤
p,
η
≤
λ,
µ
≤
κ,
ϕ
≤
q
}
=
{
ξ
r
≤
α,
η
r
≤
β,
µ
r
≤
γ,
ϕ
r
≤
θ,
ξ
≤
p,
η
≤
λ,
µ
≤
κ,
ϕ
≤
q
}
∪
{
ξ
r
≤
α,
η
r
>
β,
µ
r
γ,
ϕ
r
>
θ,
ξ
≤
p,
η
≤
λ,
µ
≤
κ,
ϕ
≤
q
}
∪
{
ξ
r
>
α,
η
r
≤
β,
µ
r
≤
γ,
ϕ
r
≤
θ,
ξ
≤
p,
η
≤
λ,
µ
≤
κ,
ϕ
≤
q
}
∪
{
ξ
r
≥
α,
η
r
≥
β,
µ
r
≥
γ,
ϕ
r
≥
θ,
ξ
≤
p,
η
≤
λ,
µ
≤
κ,
ϕ
≤
q
}
⊂
{
ξ
r
≤
α,
η
r
≤
β,
µ
r
≤
γ,
ϕ
r
≤
θ
}
∪
|
ξ
r
−
ξ
|
≥
α
−
p
∪
{|
η
r
−
η
|
≥
β
−
λ
∪
|
µ
r
−
µ
|
≥
γ
−
κ
|
∪
|
ϕ
r
−
ϕ
|
≥
θ
−
q
}
implies
D
(
p
+
i
1
λ
+
i
2
κ
+
i
1
i
2
q
)
≤
D
r
(
α
+
i
1
β
+
i
2
γ
+
i
1
i
2
θ
)
+
N
(
|
ξ
r
−
ξ
|
≥
α
−
p
)
+
N
(
|
η
r
−
η
|
≥
β
−
λ
)
+
N
(
|
µ
r
−
µ
|
≥
γ
−
κ
}
+
N
(
|
ϕ
r
−
ϕ
|
≥
θ
−
q
)
.
Thus
lim
r
→∞
N
(
|
ξ
r
−
ξ
|
≥
α
−
p
)
=
0
,
lim
r
→∞
N
(
|
η
r
−
η
|
≥
β
−
λ
)
=
0
,
lim
r
→∞
N
(
|
µ
r
−
µ
|
≥
γ
−
κ
)
=
0
,
Difference
Sequences
of
Bi-complex
Uncertain
Sequences
34
and
lim
r
→∞
N
(
|
ϕ
r
−
ϕ
|
≥
θ
−
q
)
=
0
,
we
get
D
(
p
+
i
1
λ
+
i
2
κ
+
i
1
i
2
q
)
≤
lim
r
→∞
inf
D
r
(
α
+
i
1
β
+
i
2
γ
+
i
1
i
2
θ
)
,
for
any
p
<
α,
λ
<
β,
κ
<
γ,
q
<
θ
and
p
+
i
1
λ
+
i
2
κ
+
i
1
i
2
q
→
α
+
i
1
β
+
i
2
γ
+
i
1
i
2
θ,
we
get
D
(
BC
)
≤
lim
r
→∞
inf
D
r
(
BC
)
.
(2)
From
(1)
and
(2)
,
D
r
(
BC
)
→
D
(
BC
)
as
r
→
∞
.
Thus,
(
ϑ
r
)
BC
→
ϑ
=
ξ
+
i
1
η
+
i
2
µ
+
i
1
i
2
ϕ
converges
in
distribution,
which
proves
the
theorem.
5
Conclusion
This
study
develops
a
new
framework
of
convergent
difference
sequence
spaces
involving
bi-complex
uncertain
variables
by
combining
difference
op-
erators
with
uncertainty
theory.
We
established
their
fundamental
algebraic
and
topological
properties,
including
linearity
and
normed
space
structures,
and
investigated
the
relationships
among
these
newly
defined
spaces.
In
particular,
we
proved
the
strict
inclusion
∆
n
−
1
⊂
∆
n
.
Furthermore,
we
ex-
amine
their
structural
properties
and
show
that,
in
general,
these
spaces
are
neither
solid,
monotone,
nor
symmetric.
We
also
investigated
the
conver-
gence
behavior
of
sequences
in
these
spaces
and
established
the
implications
among
different
modes
of
convergence.
The
results
presented
in
this
paper
provide
a
foundation
for
further
re-
search
on
sequence
spaces
involving
uncertain
and
bi-complex-valued
struc-
tures.
Future
work
may
focus
on
study
of
the
bi-complex
uncertain
se-
quences
defined
by
Orlicz
functions
and
extending
the
present
framework
to
statistical
convergence
and
related
summability
methods
in
the
bi-complex
uncertain
setting.
D.
Kalawathi,
B.C.
Tripathy
35
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