In this paper, we generalize and extend the Baskakov-Kantorovich operators by constructing the $(p, q)$
(
p
,
q
)
-Baskakov Kantorovich operators $$ \begin{aligned} (\Upsilon _{n,b,p,q} h) (x) = [ n ]_{p,q} \sum_{b=0}^{ \infty}q^{b-1} \upsilon _{b,n}^{p,q}(x) \int _{\mathbb{R}}h(y)\Psi \biggl( [ n ] _{p,q} \frac{q^{b-1}}{p^{n-1}}y - [ b ] _{p,q} \biggr) \,d_{p,q}y. \end{aligned} $$
(
ϒ
n
,
b
,
p
,
q
h
)
(
x
)
=
[
n
]
p
,
q
∑
b
=
0
∞
q
b
−
1
υ
b
,
n
p
,
q
(
x
)
∫
R
h
(
y
)
Ψ
(
[
n
]
p
,
q
q
b
−
1
p
n
−
1
y
−
[
b
]
p
,
q
)
d
p
,
q
y
.
The modified Kantorovich $(p, q)$
(
p
,
q
)
-Baskakov operators do not generalize the Kantorovich q-Baskakov operators. Thus, we introduce a new form of this operator. We also introduce the following useful conditions, that is, for any $0 \leq b \leq \omega $
0
≤
b
≤
ω
, such that $\omega \in \mathbb{N}$
ω
∈
N
, $\Psi _{\omega}$
Ψ
ω
is a continuous derivative function, and $0< q< p \leq 1$
0
<
q
<
p
≤
1
, we have $\int _{\mathbb{R}}x^{b}\Psi _{\omega}(x)\,d_{p,q}x = 0 $
∫
R
x
b
Ψ
ω
(
x
)
d
p
,
q
x
=
0
. Also, for every $\Psi \in L_{\infty}$
Ψ
∈
L
∞
,
there exists a finite constant γ such that $\gamma > 0$
γ
>
0
with the property $\Psi \subset [ 0, \gamma ] $
Ψ
⊂
[
0
,
γ
]
,
its first ω moment vanishes, that is, for $1 \leq b \leq \omega $
1
≤
b
≤
ω
, we have that $\int _{\mathbb{R}}y^{b}\Psi (y)\,d_{p,q}y = 0$
∫
R
y
b
Ψ
(
y
)
d
p
,
q
y
=
0
,
and $\int _{\mathbb{R}}\Psi (y)\,d_{p,q}y = 1$
∫
R
Ψ
(
y
)
d
p
,
q
y
=
1
.
Furthermore, we estimate the moments and norm of the new operators. And finally, we give an upper bound for the operator’s norm.