In this paper, we generalize and extend the Baskakov-Kantorovich operators by constructing the (p,q)\documentclass[12pt]{minimal}
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\begin{document}$(p, q)$\end{document}-Baskakov Kantorovich operators (ϒn,b,p,qh)(x)=[n]p,q∑b=0∞qb−1υb,np,q(x)∫Rh(y)Ψ([n]p,qqb−1pn−1y−[b]p,q)dp,qy.\documentclass[12pt]{minimal}
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\begin{document}$$ \begin{aligned} (\Upsilon _{n,b,p,q} h) (x) = [ n ]_{p,q} \sum_{b=0}^{ \infty}q^{b-1} \upsilon _{b,n}^{p,q}(x) \int _{\mathbb{R}}h(y)\Psi \biggl( [ n ] _{p,q} \frac{q^{b-1}}{p^{n-1}}y - [ b ] _{p,q} \biggr) \,d_{p,q}y. \end{aligned} $$\end{document} The modified Kantorovich (p,q)\documentclass[12pt]{minimal}
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\begin{document}$(p, q)$\end{document}-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≤b≤ω\documentclass[12pt]{minimal}
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\begin{document}$0 \leq b \leq \omega $\end{document}, such that ω∈N\documentclass[12pt]{minimal}
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\begin{document}$\omega \in \mathbb{N}$\end{document}, Ψω\documentclass[12pt]{minimal}
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\begin{document}$\Psi _{\omega}$\end{document} is a continuous derivative function, and 0<q<p≤1\documentclass[12pt]{minimal}
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\begin{document}$0< q< p \leq 1$\end{document}, we have ∫RxbΨω(x)dp,qx=0\documentclass[12pt]{minimal}
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\begin{document}$\int _{\mathbb{R}}x^{b}\Psi _{\omega}(x)\,d_{p,q}x = 0 $\end{document}. Also, for every Ψ∈L∞\documentclass[12pt]{minimal}
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\begin{document}$\Psi \in L_{\infty}$\end{document},there exists a finite constant γ such that γ>0\documentclass[12pt]{minimal}
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\begin{document}$\gamma > 0$\end{document} with the property Ψ⊂[0,γ]\documentclass[12pt]{minimal}
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\begin{document}$\Psi \subset [ 0, \gamma ] $\end{document},its first ω moment vanishes, that is, for 1≤b≤ω\documentclass[12pt]{minimal}
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\begin{document}$1 \leq b \leq \omega $\end{document}, we have that ∫RybΨ(y)dp,qy=0\documentclass[12pt]{minimal}
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\begin{document}$\int _{\mathbb{R}}y^{b}\Psi (y)\,d_{p,q}y = 0$\end{document},and ∫RΨ(y)dp,qy=1\documentclass[12pt]{minimal}
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\begin{document}$\int _{\mathbb{R}}\Psi (y)\,d_{p,q}y = 1$\end{document}. Furthermore, we estimate the moments and norm of the new operators. And finally, we give an upper bound for the operator’s norm.
