Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
EXPLORING
SOME
APPROXIMATION
PROCESSES
IN
QUANTUM
CALCULUS
∗
Octavian
Agratini
†
Mihaela-Diana
Lupea
‡
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.185
Abstract
The
paper
is
a
synthesis
on
a
topic
that
concerns
Approximation
Theory.
We
refer
to
the
construction
of
some
approximation
processes
using
q
-Calculus.
At
first
we
recall
the
fundamental
elements
of
q
-
Calculus
and
then
we
focus
on
the
presentation
of
special
discrete-type
operators
sequences
using
this
theory.
The
main
characteristics
of
the
new
constructions
are
highlighted.
Keywords:
positive
linear
operators,
q
-Calculus,
rate
of
convergence,
approximation
process.
MSC:
41A36,
41A25,
26D15.
1
Introduction
Quantum
Calculus
appeared
as
a
connection
between
mathematics
and
physics
having
the
characteristic
of
describing
nature
more
accurately.
It
is
known
that
the
universe
is
quantized
and
if
we
are
at
a
point
x
,
then
the
points
that
we
can
reach
are
in
geometric
progression
q
i
x
,
in
accordance
with
the
astronomer
Edwin
Hubble.
∗
Accepted
for
publication
on
July
29,
2026
†
agratini@ictp.acad.ro
,
Tiberiu
Popoviciu
Institute
of
Numerical
Analysis,
Roma-
nian
Academy,
57
Fntnele
Street,
400320
Cluj-Napoca,
Romania
‡
mihaela.lupea19@gmail.com
,
Liceul
Teoretic
Pavel
Dan
Cˆ
ampia
Turzii,
1
Decembrie
1918
Street,
no
17,
405100,
Cˆ
ampia
Turzii,
Romania
167
Exploring
some
approximation
processes
in
quantum
calculus
168
There
is
no
general
definition
of
a
”
q
-analogue”.
A
q
-analogue,
also
called
q
-extension
of
a
mathematical
object
X
,
is
a
family
of
objects
X
(
q
),
q
>
0,
in
general
q
∈
(0
,
1)
such
that
lim
q
→
1
X
(
q
)
=
X
.
In
the
last
decades,
Quantum
Calculus
began
to
be
widely
used
in
the
construction
of
linear
positive
approximation
processes.
This
work
being
a
synthesis,
after
presenting
the
fundamental
elements
regarding
q
-Calculus
in
order
to
make
the
material
essentially
self-contained,
the
authors
pursued
two
distinct
aspects:
indicating
the
representative
re-
sults
in
this
field
established
by
various
specialists
along
with
highlighting
the
authors’
achievements
published
in
time
or
newly
developed.
The
archi-
tecture
of
the
work
was
created
by
the
second
author.
Certainly,
a
work
of
this
size
does
not
include
an
exhaustive
approach.
Our
goal
is
only
to
offer
readers
a
brief
presentation
of
the
application
of
q
-Calculus
in
approximating
the
signals
belonging
to
different
spaces
of
func-
tions
with
the
help
of
linear
and
positive
operators.
We
mention
that
we
will
refer
only
to
a
few
mathematicians
who
have
made
contributions
in
this
field,
the
selection
being
subjective
therefore
random.
Among
the
pioneers
of
this
research
direction
we
mention
A.
Lupa¸
s,
G.M.
Phillips
and
V.
Gupta.
The
article
provides
the
reader
with
a
mathematical
background
and
a
wide
variety
of
examples
for
key
results.
2
Elements
of
q
-Calculus
For
the
reader’s
convenience,
we
recall
basic
definitions,
notations
and
re-
sults
regarding
Quantum
Calculus.
In
this
direction,
the
book
of
Kac
and
Cheung
can
be
consulted
[
11
].
Let
q
>
0.
For
any
n
∈
N
0
=
{
0
}
∪
N
,
the
q
-integer
[
n
]
q
and
the
q
-factorial
[
n
]
q
!
are
respectively
defined
by
[
n
]
q
=
n
−
1
j
=0
q
j
,
[
n
]
q
!
=
n
j
=1
[
j
]
q
,
n
∈
N
,
and
[0]
q
=
0,
[0]
q
!
=
1.
The
q
-binomial
coefficients
are
denoted
by
n
k
q
and
are
defined
by
n
k
q
=
[
n
]
q
!
[
k
]
q
![
n
−
k
]
q
!
,
k
=
0
,
1
,
.
.
.
,
n.
O.
Agratini,
M.-D.
Lupea
169
Clearly,
for
q
=
1
one
has
[
n
]
1
=
n
,
[
n
]
1
!
=
n
!
and
n
k
q
=
n
k
,
the
ordinary
binomial
coefficients.
In
the
sequel,
we
always
will
assume
that
q
∈
(0
,
1).
Each
q
-binomial
coefficient
is
a
polynomial
in
q
of
degree
k
(
n
−
k
)
with
1
as
the
leading
coefficient.
Also,
the
q
-binomial
coefficients
satisfy
the
following
recurrence
relations
(q-Pascal
rule):
n
+
1
k
q
=
q
n
−
k
+1
n
k
−
1
q
+
n
k
q
,
n
+
1
k
q
=
n
k
−
1
q
+
q
k
n
k
q
,
(1)
where
1
≤
k
≤
n
.
These
identities
can
be
easily
obtained
using
an
elemen-
tary
calculus,
see
for
example
[
11
,
pp.
17–18].
Definition
1.
Let
q
>
0
,
q
=
1
,
be
a
real
number.
The
q
-derivative
of
a
function
f
:
R
→
R
is
defined
by
D
q
f
(
x
)
=
f
(
x
)
−
f
(
qx
)
(1
−
q
)
x
,
x
=
0
,
D
q
f
(0)
=
lim
x
→
0
D
q
f
(
x
)
.
Definition
2.
The
function
F
is
a
q
-antiderivative
of
f
if
D
q
F
=
f
.
It
is
denoted
by
F
(
x
)
=
f
(
x
)
d
q
(
x
)
.
In
the
ordinary
calculus,
the
uniqueness
is
up
to
adding
a
constant
since
the
derivative
of
a
function
vanishes
if
and
only
if
it
is
constant.
The
sit-
uation
in
Quantum
Calculus
is
more
subtle.
D
q
ϕ
(
x
)
=
0
if
and
only
if
ϕ
(
qx
)
=
ϕ
(
x
),
which
does
not
necessarily
imply
ϕ
a
constant.
Definition
3.
Let
b
>
0
,
q
∈
(0
,
1)
and
f
:
R
→
R
.
The
real
number
denoted
by
I
(
f
;
q
;
b
)
and
defined
by
I
(
f
;
q
;
b
)
:=
b
0
f
(
x
)
d
q
x
=
(1
−
q
)
∞
k
=0
bq
k
f
(
bq
k
)
(2)
is
called
the
finite
q
-integral
of
the
function
f
on
[0
,
b
]
.
On
a
general
interval
[
a,
b
]
,
0
<
a
<
b
,
one
defines
Exploring
some
approximation
processes
in
quantum
calculus
170
b
a
f
(
t
)
d
q
t
=
b
0
f
(
t
)
d
q
t
−
a
0
f
(
t
)
d
q
t.
If
f
is
Riemann
integrable
on
[0
,
b
]
,
then
b
0
f
(
x
)
dx
=
lim
q
→
1
−
b
0
f
(
x
)
d
q
x.
Moreover,
the
q
-integral
defined
by
(
2
)
satisfies
the
relation
D
q
x
0
f
(
t
)
d
q
t
=
f
(
x
)
,
x
=
0
.
Recalling
that
for
any
r
∈
N
,
the
r
th
order
difference
operator
of
a
function
f
is
recursively
defined
by
∆
r
q
f
k
=
∆
r
−
1
q
f
k
+1
−
q
r
−
1
∆
r
−
1
q
f
k
,
where
∆
0
q
f
k
=
f
k
and
f
k
:=
f
([
k
]
q
/
[
n
]
q
),
k,
n
∈
N
,
we
present
the
connection
between
q
-difference
∆
r
q
f
0
and
r
th
derivative
f
(
r
)
∆
r
q
f
0
q
r
(
r
−
1)
[
r
]
q
!
=
f
(
r
)
(
ξ
)
r
!
,
ξ
∈
[0
,
[
r
]
q
)
,
see
[
15
,
p.268].
This
proves
the
existence
of
a
bridge
between
q
-Calculus
and
classical
analysis.
3
An
extension
of
q
-Stirling
numbers
of
the
second
kind
In
this
section,
we
present
a
new
generalization
of
Stirling
numbers
based
on
Quantum
Calculus.
Following
[
3
],
for
each
(
m,
r
)
∈
N
0
×
N
0
,
the
number
σ
q
(
m,
r
)
defined
by
σ
q
(
m,
r
)
=
1
[
r
]
q
!
r
j
=0
(
−
1)
j
q
(
r
−
j
)(
r
−
j
−
1)
/
2
r
j
q
[
r
−
j
]
m
q
q
(
r
−
j
)
m
,
(3)
represents
a
new
q
-analogue
of
Stirling
number
of
the
second
kind.
O.
Agratini,
M.-D.
Lupea
171
Theorem
1.
The
numbers
σ
q
(
m,
r
)
,
(
m,
r
)
∈
N
0
×
N
0
,
given
by
(
3
)
enjoy
the
following
properties.
σ
q
(
m,
0)
=
0
(
m
∈
N
)
and
σ
q
(0
,
0)
=
1
,
(4)
q
r
σ
q
(
m
+
1
,
r
)
=
[
r
]
q
σ
q
(
m,
r
)
+
σ
q
(
m,
r
−
1)
,
m
∈
N
0
,
r
∈
N
,
(5)
σ
q
(
m,
r
)
=
0
,
r
>
m.
(6)
Proof.
The
identity
(
4
)
follows
immediately
from
(
3
).
By
using
the
same
definition,
for
each
m
∈
N
0
,
r
∈
N
,
we
can
write
σ
q
(
m,
r
−
1)
=
1
[
r
−
1]
q
!
r
j
=1
(
−
1)
j
−
1
q
(
r
−
j
)(
r
−
j
−
1)
/
2
r
−
1
j
−
1
q
[
r
−
j
]
m
q
q
(
r
−
j
)
m
.
(7)
On
the
other
hand,
q
r
σ
q
(
m
+
1
,
r
)
−
[
r
]
q
σ
q
(
m,
r
)
=
[
r
]
q
[
r
]
q
!
q
r
−
1
r
j
=0
(
−
1)
j
q
(
r
−
j
−
1)(
r
−
j
−
2)
/
2
r
−
1
j
q
[
r
−
j
]
m
q
q
(
r
−
j
)
m
−
[
r
]
q
[
r
]
q
!
r
j
=0
(
−
1)
j
q
(
r
−
j
)(
r
−
j
−
1)
/
2
r
j
q
[
r
−
j
]
m
q
q
(
r
−
j
)
m
=
1
[
r
−
1]
q
!
r
j
=0
(
−
1)
j
q
(
r
−
j
−
1)(
r
−
j
−
2)
/
2
[
r
−
j
]
m
q
q
(
r
−
j
)
m
q
r
−
1
r
−
1
j
q
−
q
r
−
j
−
1
r
j
q
=
1
[
r
−
1]
q
!
r
j
=0
(
−
1)
j
q
(
r
−
j
−
1)(
r
−
j
−
2)
/
2
[
r
−
j
]
m
q
q
(
r
−
j
)
m
q
r
−
1
r
−
1
j
q
−
1
q
j
r
j
q
.
Taking
into
account
(
1
)
with
n
:=
r
−
1,
k
:=
j
and
(
7
)
the
proof
is
complete.
For
establishing
relation
(
6
),
on
the
basis
of
(
5
),
it
is
enough
to
prove
σ
q
(
m,
m
+
1)
=
0
for
any
m
∈
N
0
.
Since
[
m
+
1
−
j
]
m
q
=
1
(1
−
q
)
m
m
k
=0
(
−
1)
k
m
k
q
k
(
m
+1)
−
kj
,
for
0
≤
j
≤
m
+
1
,
and
m
+1
j
=0
q
j
(
j
−
1)
/
2
m
+
1
j
q
−
1
q
k
j
=
1
−
1
q
k
m
+1
q
=
0
,
for
0
≤
k
<
m
+1
,
Exploring
some
approximation
processes
in
quantum
calculus
172
we
can
write
σ
q
(
m,
m
+1)
=
1
[
m
+
1]
q
!
q
m
(
m
+1)
/
2
m
+1
j
=0
(
−
1)
j
j
j
(
j
−
1)
/
2
m
+
1
j
q
[
m
+1
−
j
]
m
q
=
(1
−
q
)
−
m
[
m
+
1]
q
!
q
m
(
m
+1)
/
2
m
k
=0
(
−
1)
k
m
k
q
k
(
m
+1)
1
−
1
q
k
m
+1
q
=
0
.
Easily
it
can
be
proved
lim
q
→
1
−
σ
q
(
m,
r
)
=
1
r
!
r
j
=0
(
−
1)
r
−
j
r
j
j
m
,
m
∈
N
0
,
r
∈
N
0
,
the
limit
representing
S
(
m,
r
),
the
classical
Stirling
numbers
of
the
second
kind,
see
[
1
,
p.824].
It
is
known
that
a
q
-analogue
of
Stirling
numbers
is
not
unique.
For
ex-
ample
Aral
[
5
]
introduced
the
numbers
S
q
(
m,
r
)
in
studying
a
q
-generalization
of
Sz´
asz–Mirakjan
operators.
Further,
we
focus
on
using
q
-Calculus
to
obtain
sequences
of
linear
ap-
proximation
operators.
The
first
researches
have
been
achieved
by
Lupa¸
s
[
12
]
and
Phillips
[
16
]
who
proposed
q
-variants
of
the
original
Bernstein
operators.
Also,
along
the
way,
other
classes
of
operators
have
been
extended,
for
ex-
ample:
Meyer–K¨
onig
and
Zeller
operators
[
19
],
Bleimann,
Butzer
and
Hahn
operators
[
6
],
Sz´
asz–Mirakjan
operators
[
5
],
Bal´
azs–Szabados
operators
[
8
].
4
A
class
of
Stancu
operators
in
q
-Calculus
For
f
∈
C
([0
,
1]),
α
≥
0
and
each
n
∈
N
,
Nowak
[
14
]
have
been
defined
the
operators
(
B
q,α
n
f
)(
x
)
=
n
k
=0
p
q,α
n,k
(
x
)
f
[
k
]
q
[
n
]
q
,
x
∈
[0
,
1]
,
(8)
where
p
q,α
n,k
(
x
)
=
n
k
q
k
−
1
i
=0
(
x
+
α
[
i
]
q
)
n
−
1
−
k
s
=0
(1
−
q
s
x
+
α
[
s
]
q
)
n
−
1
i
=0
(1
+
α
[
i
]
q
)
.
O.
Agratini,
M.-D.
Lupea
173
We
mention,
an
empty
product
is
taken
to
be
equal
to
1.
This
class
contains
as
special
cases
to
following
well-known
sequences
of
linear
and
positive
operators:
(i)
For
α
=
0,
B
q,
0
n
≡
B
q
n
represents
q
-Bernstein
operator
introduced
by
Phillips.
(ii)
For
q
=
1
and
α
=
0,
B
1
,
0
n
≡
B
n
is
the
classical
Bernstein
polynomial.
(iii)
For
q
=
1,
B
1
,α
n
≡
B
α
n
turns
into
Stancu
operator
defined
as
follows
(
B
α
n
f
)(
x
)
=
1
1
[
n,
−
α
]
n
k
=0
n
k
x
[
k,
−
α
]
(1
−
x
)
[
n
−
k,
−
α
]
f
k
n
,
x
∈
[0
,
1]
.
Here,
t
[
m,α
]
=
m
−
1
j
=0
(
t
−
jα
)
represents
the
generalized
factorial
power
with
the
step
α
∈
R
,
m
∈
N
.
In
[
2
]
the
author
proved
the
following
identities
(
B
q,α
n
e
0
)(
x
)
=
1
,
(
B
q,α
n
e
1
)(
x
)
=
x,
(
B
q,α
n
e
2
)(
x
)
=
1
1
+
α
x
(
x
+
α
)
+
x
(1
−
x
)
[
n
]
q
,
x
∈
[0
,
1]
,
(9)
where
e
j
,
j
∈
N
,
stands
for
the
monomial
of
j
-th
degree
and
e
0
(
x
)
=
1.
On
the
basis
of
Bohman–Korovkin
theorem,
the
conclusion
lim
n
→∞
(
B
q
n
,α
n
n
f
)(
x
)
=
f
(
x
)
uniformly
in
x
∈
[0
,
1]
,
(10)
takes
place
with
additional
hypotheses
0
<
q
n
<
1
,
lim
n
→∞
q
n
=
1
and
α
n
≥
0
,
lim
n
→∞
α
n
=
0
.
(11)
The
operator
B
q,α
n
can
be
reintroduced
by
using
probabilistic
tools,
see
[
2
].
Let
(Ω
,
F
,
P
)
be
a
probability
space
and
Z
q
:
N
×
[0
,
1]
→
M
2
(Ω)
a
ran-
dom
scheme
on
[0
,
1],
where
M
2
(Ω)
stands
for
the
space
of
all
real
square-
integrable
random
variables
on
Ω.
We
consider
P
Z
q
(
n,
x
)
=
[
k
]
q
[
n
]
q
=
p
q,α
n,k
(
x
)
,
0
≤
k
≤
n,
n
∈
N
,
x
∈
[0
,
1]
.
Exploring
some
approximation
processes
in
quantum
calculus
174
As
usual,
the
mathematical
expectation
and
the
variance
of
Z
q
(
n,
x
)
are
denoted
by
E
(
Z
q
(
n,
x
))
and
Var(
Z
q
(
n,
x
)),
respectively.
Setting
P
Z
q
(
n,x
)
the
distribution
of
Z
q
(
n,
x
)
with
respect
to
P
,
one
has
(
B
q,α
n
f
)(
x
)
=
Ω
f
◦
Z
q
(
n,
x
)
dP
=
1
0
f
dP
Z
q
(
n,x
)
=
E
f
(
Z
q
(
n,
x
))
,
(12)
x
∈
[0
,
1]
and,
in
harmony
with
(
9
),
we
obtain
E
(
Z
q
(
n,
x
))
=
x,
Var(
Z
q
(
n,
x
))
=
1
+
α
[
n
]
q
(1
+
α
)[
n
]
q
x
(1
−
x
)
.
(13)
Theorem
2.
Under
the
hypotheses
(
11
)
,
identity
(
10
)
holds.
Proof.
(probabilistic
approach).
For
proving
the
statement,
it
is
enough
to
show
the
following.
For
any
ε
>
0,
there
exist
η
ε
>
0
and
n
ε
∈
N
such
that
|
(
B
q
n
,α
n
n
f
)(
x
)
−
f
(
x
)
|
<
ε,
(14)
for
all
x
∈
[0
,
1],
q
n
∈
(1
−
η
ε
,
1),
α
n
∈
(0
,
η
ε
)
and
n
≥
n
ε
.
Let
ε
>
0
be
arbitrarily
fixed.
Since
f
∈
C
([0
,
1]),
a
constant
M
exists
such
that
|
f
(
x
)
|
≤
M
for
all
x
∈
[0
,
1].
Let
δ
∈
(0
,
1]
be
chosen
such
that
|
f
(
t
)
−
f
(
x
)
|
<
ε/
2,
whenever
|
t
−
x
|
<
δ
,
t
∈
[0
,
1],
x
∈
[0
,
1].
For
all
t
and
x
belonging
to
[0
,
1]
such
that
|
t
−
x
|
≥
δ
,
we
can
write
|
f
(
t
)
−
f
(
x
)
|
≤
2
M
≤
2
M
(
t
−
x
)
2
δ
2
.
Let
η
ε
:=
εδ
2
/M
,
where
δ
≤
1.
The
constant
M
can
be
taken
as
large
as
we
want,
consequently
η
ε
<
1.
On
the
other
hand,
since
lim
n
→∞
α
n
=
0,
lim
n
→∞
q
n
=
1,
we
get
lim
n
→∞
[
n
]
q
n
=
∞
and,
for
the
above
chosen
η
ε
,
n
ε
∈
N
exists
such
that
α
n
+
1
[
n
]
q
n
<
η
ε
for
each
n
≥
n
ε
.
(15)
Taking
in
view
both
(
12
),
(
13
)
and
(
15
)
we
can
write
successively
|
(
B
q
n
,α
n
n
f
)(
x
)
−
f
(
x
)
|
≤
1
0
|
f
(
t
)
−
f
(
x
)
|
P
Z
q
n
(
n,x
)
(
dt
)
=
t
∈
[0
,
1]
|
t
−
x
|
<δ
+
t
∈
[0
,
1]
|
t
−
x
|≥
δ
≤
ε
2
1
0
dP
Z
q
n
(
n,x
)
+
2
M
δ
2
1
0
(
t
−
x
)
2
dP
Z
q
n
(
n,x
)
=
ε
2
+
2
M
δ
2
Var(
Z
q
n
(
n,
x
))
≤
ε
2
+
2
M
δ
2
α
n
+
1
[
n
]
q
n
max
x
∈
[0
,
1]
x
(1
−
x
)
<
ε.
Therefore,
(
14
)
is
true
and
the
proof
is
ended.
O.
Agratini,
M.-D.
Lupea
175
Further
we
explore
the
rate
of
convergence
of
B
q,α
n
,
0
<
q
<
1,
α
>
0,
n
∈
N
,
in
terms
of
the
modulus
of
continuity
ω
1
(
f
;
·
),
where
ω
1
(
f
;
δ
)
=
sup
x,y
∈
[0
,
1]
|
x
−
y
|≤
δ
|
f
(
x
)
−
f
(
y
)
|
,
δ
≥
0
.
Theorem
3.
Let
B
q,α
n
,
n
∈
N
,
be
defined
by
(
8
)
.
(i)
If
f
∈
C
([0
,
1])
,
then
|
(
B
q,α
n
f
)(
x
)
−
f
(
x
)
|
≤
3
2
ω
1
f
;
1
+
α
[
n
]
q
(1
+
α
)[
n
]
q
,
x
∈
[0
,
1]
.
(ii)
If
f
is
differentiable
and
f
∈
C
([0
,
1])
,
then
|
(
B
q,α
n
f
)(
x
)
−
f
(
x
)
|
≤
3
4
1
+
α
[
n
]
q
(1
+
α
)[
n
]
q
ω
1
f
;
1
+
α
[
n
]
q
(1
+
α
)[
n
]
q
,
x
∈
[0
,
1]
.
Proof.
(i)
Based
on
a
result
due
to
Shisha
and
Mond
[
18
],
for
any
linear
positive
operator
L
:
C
([0
,
1])
→
B
([0
,
1])
the
following
inequality
involving
the
modulus
of
continuity,
takes
place
|
(
Lf
)(
x
)
−
f
(
x
)
|
≤
|
f
(
x
)
|
|
(
Le
0
)(
x
)
−
1
|
(16)
+
(
Le
0
)(
x
)
+
1
δ
(
Lϕ
2
n
)(
x
)
(
Le
0
)(
x
)
ω
1
(
f
;
δ
)
,
x
∈
[0
,
1],
δ
>
0,
where
ϕ
x
(
t
)
=
t
−
x
,
t
∈
[0
,
1].
In
view
of
(
9
)
we
obtain
(
B
q,α
n
ϕ
2
x
)(
x
)
=
1
+
α
[
n
]
q
(1
+
α
)[
n
]
q
x
(1
−
x
)
.
Since
x
(1
−
x
)
≤
1
4
,
x
∈
[0
,
1],
choosing
δ
=
1
+
α
[
n
]
q
(1
+
α
)[
n
]
q
,
the
conclusion
follows.
(ii)
Under
the
assumption
f
∈
C
([0
,
1])
one
has
[
18
]
|
(
Lf
)(
x
)
−
f
(
x
)
|
≤
|
f
(
x
)
|
|
(
Le
0
)(
x
)
−
1
|
+
|
f
(
x
)
|
|
(
Lϕ
x
)(
x
)
|
+
(
Lϕ
2
x
)(
x
)
(
Le
0
)(
x
)
+
1
δ
(
Lϕ
2
x
)(
x
)
ω
1
(
f
;
δ
)
.
Since
(
B
q,α
n
ϕ
x
)(
x
)
=
0
with
the
same
choice
of
δ
,
the
conclusion
follows.
Exploring
some
approximation
processes
in
quantum
calculus
176
By
definition
the
m
-th
iterate
of
B
q,α
n
is
1
B
q,α
n
:=
B
q,α
n
,
m
B
q,α
n
:=
B
q,α
n
(
m
−
1)
B
q,α
n
,
m
=
2
,
3
,
.
.
.
Our
next
aim
is
to
study
the
convergence
of
the
iterates
m
B
q,α
n
as
m
tends
to
infinity.
To
achieve
this,
we
recall
a
result
obtained
by
using
the
contraction
principle.
Theorem
4.
(
[
4
])
Let
L
n
,
n
∈
N
,
be
defined
as
follows
L
n
:
C
([
a,
b
])
→
C
([
a,
b
])
,
(
L
n
f
)(
x
)
=
n
k
=0
ψ
n,k
(
x
)
f
(
x
n,k
)
,
where
0
=
x
n,
0
<
x
n,
1
<
·
·
·
<
x
n,n
=
b
and
for
each
0
≤
k
≤
n
,
ψ
n,k
∈
C
([
a,
b
])
.
We
assume
that
L
n
e
j
=
e
j
,
j
∈
{
0
,
1
}
.
Let
us
denote
u
n
=
min
x
∈
[
a,b
]
ψ
n,
0
(
x
)
+
ψ
n,n
(
x
)
.
If
u
n
>
0
,
then
lim
m
→∞
m
L
n
f
(
x
)
=
f
(
a
)
+
f
(
b
)
−
f
(
a
)
b
−
a
(
x
−
a
)
,
f
∈
C
([
a,
b
])
,
uniformly
on
[
a,
b
]
.
Theorem
5.
Let
B
q,α
n
,
n
∈
N
,
be
defined
by
(
8
)
.
For
any
fixed
n
∈
N
,
one
has
lim
m
→∞
m
B
q,α
n
(
x
)
=
f
(0)
+
f
(1)
−
f
(0)
x,
f
∈
C
([0
,
1])
,
uniformly
on
[0
,
1]
.
Proof.
Choosing
in
Theorem
4.3
a
=
0,
b
=
1,
x
n,k
=
[
k
]
q
/
[
n
]
q
,
ψ
n,k
=
p
q,α
n,k
,
0
≤
k
≤
n
,
and
knowing
the
identities
(
9
),
all
is
left
to
be
proved
is
the
relation
min
x
∈
[0
,
1]
p
q,α
n,
0
(
x
)
+
p
q,α
n,n
(
x
)
>
0
.
O.
Agratini,
M.-D.
Lupea
177
We
get
p
q,α
n,
0
(
x
)
+
p
q,α
n,n
(
x
)
=
n
−
1
i
=0
(1
+
α
[
i
]
q
)
−
1
n
−
1
s
=0
(
x
+
α
[
s
]
q
)
+
n
−
1
s
=0
(1
−
q
s
x
+
α
[
s
]
q
)
≥
n
−
1
i
=0
(1
+
α
[
i
]
q
)
−
1
x
n
+
(1
−
x
)
n
≥
1
2
n
−
1
n
−
1
i
=0
(1
+
α
[
i
]
q
)
−
1
≥
1
2
n
−
1
1
+
α
1
−
q
−
n
,
and,
consequently,
Theorem
4.3
can
be
applied
and
the
proof
is
complete.
5
q-Operators
on
unbounded
intervals
We
propose
to
present
two
families
of
operators
that
act
on
spaces
of
func-
tions
defined
on
R
+
=
[0
,
∞
).
5.1
q-Baskakov
operators
Aral
and
Gupta
[
7
]
defined
a
q
-analogue
of
Baskakov
operators,
which
for
q
∈
(0
,
1),
f
∈
C
(
R
+
),
x
∈
R
+
,
n
∈
N
,
is
defined
by
(
V
n,q
f
)(
x
)
=
∞
k
=0
v
n,k
(
q
;
x
)
f
[
k
]
q
q
k
−
1
[
n
]
q
,
(17)
where
v
n,k
(
q
;
x
)
=
n
+
k
−
1
k
q
q
k
(
k
−
1)
2
x
k
(1
+
x
)
n
+
k
q
.
Taking
into
account
q
-Taylor
Theorem
and
the
identity
(
−
x
)
k
q
=
(
−
x
)
k
q
k
(
k
−
1)
2
,
we
get
∞
k
=0
v
n,k
(
q
;
x
)
=
1
.
Exploring
some
approximation
processes
in
quantum
calculus
178
It
is
clear
that
the
operators
given
by
(
17
)
are
positive
and
linear
oper-
ators.
For
q
=
1,
they
reduce
to
the
classical
Baskakov
operators.
In
[
10
],
the
authors
proved
V
n,q
(
e
0
;
x
)
=
1
,
V
n,q
(
e
1
;
x
)
=
x,
V
n,q
(
e
2
;
x
)
=
[
n
+
1]
q
q
[
n
]
q
x
2
+
1
[
n
]
q
x.
(18)
Based
on
q
-integration,
Gupta
and
Radu
[
10
]
also
introduced
the
Kan-
torovich
variant
of
the
above
operators
as
follows
V
K
n,q
(
f
;
x
)
=
[
n
]
q
∞
k
=0
v
n,k
(
q
;
x
)
[
k
+1]
q
[
n
]
q
q
[
k
]
q
[
n
]
q
f
(
q
−
k
+1
t
)
d
q
t,
x
∈
R
+
,
n
∈
N
.
From
Definition
2.3,
by
a
simple
computation
arising
from
(
18
)
,
the
following
identities
V
K
n,q
(
e
0
;
x
)
=
1
,
V
K
n,q
(
e
1
;
x
)
=
x
+
q
[2]
q
[
n
]
q
,
V
K
n,q
(
e
2
;
x
)
=
[
n
+
1]
q
q
[
n
]
q
x
2
+
q
(1
+
[2]
q
)
+
[3]
q
[3]
q
[
n
]
q
x
+
q
2
[3]
q
[
n
]
2
q
were
obtained.
Examining
the
last
identity,
it
is
clear
that
the
sequence
of
the
operators
(
V
K
n,q
)
n
≥
1
does
not
satisfy
the
conditions
of
Bohman-
Korovkin
theorem.
The
transformation
of
this
sequence
into
an
approxima-
tion
process
is
achieved
by
replacing
the
parameter
q
with
a
certain
sequence,
as
can
be
read
in
the
following
result.
Theorem
6.
(
[
10
])
Let
(
q
n
)
n
∈
N
be
a
sequence
satisfying
0
<
q
n
<
1
,
lim
n
→∞
q
n
=
1
.
Then,
for
any
compact
J
⊂
R
+
and
for
each
f
∈
C
(
R
+
)
,
we
have
lim
n
→∞
V
K
n,q
n
(
f
;
x
)
=
f
(
x
)
,
uniformly
in
x
∈
J
.
5.2
q
-Baskakov–Mastroianni
operators
Let
(
φ
n
)
n
≥
1
be
a
sequence
of
real
valued
functions
defined
on
R
+
,
continu-
ously
infinitely
q
-differentiable
on
R
+
,
q
∈
(0
,
1),
and
satisfying
the
following
conditions:
O.
Agratini,
M.-D.
Lupea
179
(
P
1
)
φ
n
(0)
=
1
,
n
∈
N
,
(
P
2
)
(
−
1)
k
D
k
q
φ
n
(
x
)
≥
0
,
n
∈
N
,
k
∈
N
0
,
x
≥
0
.
For
all
(
n,
k
)
∈
N
×
N
0
there
exists
a
positive
integer
i
k
,
0
≤
i
k
≤
k
,
such
that
(
P
3
)
D
k
+1
q
φ
n
(
x
)
=
(
−
1)
i
k
+1
D
k
−
i
k
q
φ
n
q
i
k
+1
x
β
n,k,i
k
,q
(
x
)
,
where
lim
n
→∞
β
n,k,i
k
,q
(0)
[
n
]
i
k
+1
q
q
k
−
i
k
=
1
.
Inspired
by
the
general
class
introduced
in
[
13
],
Radu
[
17
]
defined
the
following
operators
T
n,q
(
f
;
x
)
=
∞
k
=0
(
−
x
)
k
[
k
]
q
!
q
k
(
k
−
1)
2
D
k
q
φ
n
(
x
)
f
[
k
]
q
[
n
]
q
q
k
−
1
,
x
≥
0
,
(19)
f
∈
F
(
R
+
)
:=
{
f
:
R
+
→
R
,
the
series
in
(
19
)
is
convergent
}
,
see
also
[
3
,
Eq.
(16)].
For
each
n
∈
N
,
T
n,q
are
linear
positive
operators
satisfying
the
inter-
polating
property
T
n,q
(
f
;
0)
=
f
(0)
.
With
the
help
of
q
-Stirling
numbers
presented
at
Section
3,
the
next
result
indicates
the
moments
of
the
opera-
tors
T
n,q
,
see
[
3
,
Lemma
4].
Theorem
7.
Let
T
n,q
,
n
∈
N
,
be
defined
by
(
19
)
.
One
has
T
n,q
(
e
m
;
x
)
=
m
r
=0
(
−
x
)
r
[
n
]
m
q
q
m
D
r
q
φ
n
(0)
σ
q
(
m,
r
)
,
x
≥
0
.
(20)
Proof.
Let
f
∈
F
(
R
+
)
be
arbitrarily
fixed.
By
using
∆
0
q
f
k,s
=
f
k,s
,
f
k,s
=
f
[
k
]
q
q
s
[
n
]
q
,
k
∈
N
0
,
s
∈
Z
,
the
operator
T
n,q
can
be
expressed
as
follows
T
n,q
(
f
;
x
)
=
∞
k
=0
(
−
x
)
k
[
k
]
q
!
q
k
(
k
−
1)
2
D
k
q
φ
n
(
x
)
∆
0
q
f
k,k
−
1
.
Exploring
some
approximation
processes
in
quantum
calculus
180
On
the
basis
of
the
product
rule
in
q
-Calculus,
the
q
-derivative
of
T
n,q
f
is
given
by
D
q
T
n,q
(
f
;
x
)
=
−
∞
k
=0
(
−
x
)
k
[
k
]
q
!
q
k
(
k
+1)
2
D
k
+1
q
φ
n
(
x
)
∆
0
q
f
k
+1
,k
+
∞
k
=0
(
−
x
)
k
[
k
]
q
!
q
k
q
k
(
k
−
1)
2
D
k
+1
q
φ
n
(
x
)
∆
0
q
f
k,k
−
1
=
−
∞
k
=0
(
−
x
)
k
[
k
]
q
!
q
k
(
k
−
1)
2
q
k
D
k
+1
q
φ
n
(
x
)
∆
0
q
f
k
+1
,k
−
∆
0
q
f
k,k
−
1
=
−
∞
k
=0
(
−
x
)
k
[
k
]
q
!
q
k
(
k
−
1)
2
q
k
D
k
+1
q
φ
n
(
x
)
∆
1
q
f
k,k
.
For
n
∈
N
and
x
∈
R
+
,
by
induction
with
respect
to
r
∈
N
we
can
prove
D
r
q
T
n,q
(
f
;
x
)
=
(
−
1)
r
∞
k
=0
(
−
x
)
k
[
k
]
q
!
q
k
(
k
−
1)
2
q
rk
D
k
+
r
q
φ
n
(
x
)
∆
r
q
f
k,k
+
r
−
1
.
Choosing
x
=
0,
we
deduce
D
r
q
T
n,q
(
f
;
0)
=
(
−
1)
r
D
r
q
φ
n
(0)
σ
q
(
m,
r
)
.
Consequently,
taking
into
account
the
identity
[
3
,
Eq.
(11)],
we
obtain
∆
r
q
f
0
,r
−
1
=
q
m
[
r
]
q
!
[
n
]
m
q
σ
q
(
m,
r
)
,
r
∈
N
0
,
f
:=
e
m
,
involving
D
r
q
T
n,q
(
e
m
;
0)
=
(
−
1)
r
q
m
[
r
]
q
!
[
n
]
m
q
D
r
q
φ
n
(0)
σ
q
(
m,
r
)
.
Choosing
a
=
0
in
the
q
-Taylor
theorem
(see
[
9
,
p.
103]),
we
obtain
T
n,q
(
e
m
;
x
)
=
∞
r
=0
x
r
[
r
]
q
!
D
r
q
T
n,q
(
e
m
;
0)
=
∞
r
=0
(
−
x
)
r
[
n
]
m
q
q
m
D
r
q
φ
n
(0)
σ
q
(
m,
r
)
.
Taking
into
account
(
6
),
the
proof
is
complete.
O.
Agratini,
M.-D.
Lupea
181
Formula
(
20
)
gives
the
explicit
form
of
the
first
three
moments
of
T
n,q
,
n
∈
N
,
operators.
T
n,q
(
e
0
;
x
)
=
1
,
T
n,q
(
e
1
;
x
)
=
−
x
D
q
φ
n
(0)
[
n
]
q
,
T
n,q
(
e
2
;
x
)
=
x
2
D
2
q
φ
n
(0)
q
[
n
]
2
q
−
x
D
q
φ
n
(0)
[
n
]
2
q
.
Consequently,
the
second
central
moment
of
T
n,q
,
n
∈
N
,
is
given
as
follows
T
n,q
(
ϕ
2
x
;
x
)
=
a
n,q
x
2
+
b
n,q
x,
x
≥
0
,
where
where
a
n,q
=
1
+
2
D
q
φ
n
(0)
[
n
]
q
+
D
2
q
φ
n
(0)
q
[
n
]
2
q
,
b
n,q
=
−
D
q
φ
n
(0)
[
n
]
2
q
.
(21)
Clearly,
lim
n
→∞
T
n,q
(
e
2
;
x
)
=
x
2
.
To
transform
these
operators
into
an
ap-
proximation
process,
the
constant
q
will
be
replaced
by
a
q
n
∈
(0
,
1),
such
that
lim
n
→∞
q
n
=
1.
We
are
able
to
present
both
the
convergence
of
the
operators
and
their
rate
of
convergence.
The
results
appear
in
[
3
],
see
Theorems
1
and
2.
Theorem
8.
Let
(
q
n
)
n
≥
1
,
0
<
q
n
<
1
,
be
a
sequence
and
let
T
n,q
n
,
n
∈
N
,
be
defined
as
in
(
19
)
.
If
lim
n
→∞
q
n
=
1
,
for
any
compact
K
⊂
R
+
and
for
each
f
∈
F
(
R
+
)
∩
C
(
R
+
)
one
has
lim
n
→∞
T
n,q
n
(
f
;
x
)
=
f
(
x
)
uniformly
in
x
∈
K.
Moreover,
for
every
f
∈
C
B
(
R
+
)
one
has
|
T
n,q
n
(
f
;
x
)
−
f
(
x
)
|
≤
1
+
max
{
x,
x
2
}
ω
1
f
;
√
c
n,q
n
,
x
≥
0
,
where
c
n,q
n
=
|
a
n,q
n
|
+
b
n,q
n
,
these
sequences
being
defined
by
(
21
)
.
Proof.
The
first
statement
is
based
on
the
Bohman–Korovkin
theorem
and
the
expression
of
the
first
three
calculated
moments.
For
determine
the
convergence
speed,
we
used
relation
(
16
)
written
to
an
arbitrary
interval
[0
,
x
]
and
the
inequalities
0
≤
T
n,q
n
(
ϕ
2
x
;
x
)
≤
max
{
x,
x
2
}
c
n,q
n
.
Exploring
some
approximation
processes
in
quantum
calculus
182
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