The
\(( p,q ) \)-calculus is a generalization of the well-known
q-calculus and it is constructed by the following notations and definitions. Let
\(0< q< p\leq1\). For each nonnegative integer
n, the
\((p,q)\)-number is denoted by
\([ n ] _{p,q}\) and is given by
$$ [ n ] _{p,q}=\frac{p^{n}-q^{n}}{p-q}. $$
For each
\(k,n\in\mathbb{N}\),
\(n\geq k\geq0\), the
\((p,q)\)-factorial
\([k ] _{p,q}!\) and
\((p,q)\)-binomial are defined by
$$\begin{aligned}& [ n ] _{p,q}!=\prod_{k=1}^{n} [ k ] _{p,q},\quad n\geq 1,\qquad [ 0 ] _{p,q}!=1, \\& \left [ \textstyle\begin{array}{@{}c@{}} n \\ k\end{array}\displaystyle \right ] _{p,q}= \frac{ [ n ] _{p,q}!}{ [ n-k ] _{p,q}! [ k ] _{p,q}!}. \end{aligned}$$
The
\((p,q)\)-power basis is defined by
$$ (x\oplus a)_{p,q}^{n}=(x+a) (px+qa) \bigl(p^{2}x+q^{2}a \bigr)\cdots\bigl(p^{n-1}x+q^{n-1}a\bigr) $$
and
$$ (x\ominus a)_{p,q}^{n}=(x-a) (px-qa) \bigl(p^{2}x-q^{2}a \bigr)\cdots\bigl(p^{n-1}x-q^{n-1}a\bigr). $$
Let
\(f:\mathbb{R\rightarrow R}\) then the
\((p,q)\)-derivative of a function
f, denoted by
\(D_{p,q}f\), is defined by
$$ ( D_{p,q}f ) ( x ) :=\frac{f ( px ) -f ( qx ) }{ ( p-q ) x},\quad x\neq0,\qquad ( D_{p,q}f ) ( 0 ) :=f^{\prime} ( 0 ) $$
provided that
f is differentiable at 0. The following assertions hold true:
$$\begin{aligned}& D_{p,q}(x\oplus a)_{p,q}^{n} = [ n ] _{p,q}(px\oplus a)_{p,q}^{n-1},\quad n\geq1, \\& D_{p,q}(a\oplus x)_{p,q}^{n} = [ n ] _{p,q}(a\oplus qx)_{p,q}^{n-1},\quad n\geq1, \end{aligned}$$
and
\(D_{p,q}(x\oplus a)_{p,q}^{0}=0\). The formula for the
\((p,q)\)-derivative of a product is
$$ D_{p,q} \bigl( u ( x ) v ( x ) \bigr) :=D_{p,q} \bigl( u ( x ) \bigr) v ( qx ) +D_{p,q} \bigl( v ( x ) \bigr) u ( px ) . $$
Let
\(f:C [ 0,a ] \rightarrow \mathbb{R}\) (
\(a>0\)) then the
\((p,q)\)-integration of a function
f is defined by
$$ \begin{aligned} & \int _{0}^{a}f ( t ) \, d_{p,q}t = ( q-p ) a \sum_{k=0}^{\infty}f \biggl( \frac{p^{k}}{q^{k+1}}a \biggr) \frac{p^{k}}{q^{k+1}}\quad \text{if }\biggl\vert \frac{p}{q}\biggr\vert < 1, \\ & \int _{0}^{a}f ( t ) \, d_{p,q}t = ( p-q ) a \sum_{k=0}^{\infty}f \biggl( \frac{q^{k}}{p^{k+1}}a \biggr) \frac{q^{k}}{p^{k+1}}\quad \text{if }\biggl\vert \frac{p}{q}\biggr\vert >1. \end{aligned} $$
(1.1)
The formula of the
\((p,q)\)-integration by parts is given by
$$ \int_{a}^{b}f ( px ) D_{p,q}g ( x )\, d_{p,q}x=f ( b ) g ( b ) -f ( a ) q ( a ) - \int_{a}^{b}g ( qx ) D_{p,q}f ( x )\, d_{p,q}x. $$
(1.2)
Here we note that all the notations mentioned above reduce to the
q-analogs when
\(p=1\). For more details of the
\((p,q)\)-calculus, we refer the reader to [
1‐
5].
The
\((p,q)\)-calculus has been used efficiently in many fields of science such as oscillator algebra, Lie group, field theory, differential equations, hypergeometric series, physical sciences. Therefore, to approximate the functions via polynomials based on
\((p,q)\)-integers, no doubt, would have a crucial role. To fulfill this necessity, very recently the well-known sequences of linear positive operators of approximation theory have been transferred to the
\((p,q)\)-calculus and the advantages of
\((p,q)\) analogs of them have been intensively investigated. For some recent work devoted to
\((p,q)\)-operators, we refer the reader to [
6‐
11]. Very recently, Aral and Gupta [
12] introduced the
\(( p,q ) \)-analog of the well-known Baskakov operators by
$$ B_{n,p,q} ( f;x ) =\sum_{k=0}^{\infty}b_{n,k}^{p,q}(x)f \biggl( \frac{p^{n-1}[k]_{p,q}}{q^{k-1}[n]_{p,q}} \biggr) , $$
(1.3)
where
\(x\in{}[0,\infty)\),
\(0< q< p\leq1\), and
$$ b_{n,k}^{p,q}(x)=\left [ \textstyle\begin{array}{@{}c@{}} n+k-1 \\ k\end{array}\displaystyle \right ] _{p,q}p^{k+n(n-1)/2}q^{k(k-1)/2}\frac{x^{k}}{(1\oplus x)_{p,q}^{n+k}}, $$
and they calculated that
$$ B_{n,p,q} ( 1;x ) =1,\qquad B_{n,p,q} ( t;x ) =x,\qquad B_{n,p,q} \bigl( t^{2};x \bigr) =x^{2}+ \frac{p^{n-1}x}{[n]_{p,q}} \biggl( 1+\frac {p}{q}x \biggr) . $$
(1.4)
Another problem in the approximation by linear positive operators is to present an approximation process for Riemann integrable functions. The main tool to solve this problem is to consider the Kantorovich modifications of the corresponding operators, which mainly depends on the replacing the sample values
\(f ( k/n ) \) by the mean values of
f in the intervals
\([ k/ ( n+1 ) , ( k+1 ) / ( n+1 ) ] \). Since the
\(( p,q ) \)-integral of
f over
\([ a,b ] \) is defined as follows:
$$ \int_{a}^{b}f ( t )\, d_{p,q}t= \int_{0}^{b}f ( t )\, d_{p,q}t- \int_{0}^{a}f ( t )\, d_{p,q}t, $$
(1.5)
one cannot say (
1.5) is positive every time unless it is assumed that
f is a nondecreasing function. Hence, use of (
1.5) to introduce a Kantorovich modification of any
\(( p,q ) \)-operators may lead to some problem. Recently Mursaleen
et al. [
13] introduced a Kantorovich modification of
\(( p,q ) \)-Szász-Mirakjan operators using the
\(( p,q ) \)-integral (
1.5) for the functions being nondecreasing. However, in this paper we define a new
\(( p,q ) \)-integral, hence we do not need to impose any condition on
f. For the generalizations of Baskakov operators and Kantorovich operators in classical calculus and
q-calculus, we refer the reader to some recent papers [
14‐
19].
The aim of this paper is to introduce
\(( p,q ) \)-Baskakov-Kantorovich operators and investigate their approximation properties. In the next section, we construct the operators, calculate the moments, central moments of the operators, and give some lemmas which will be necessary to prove our main results. In Section
3, we prove a local approximation theorem for the new operators in terms of Peetre’s
\(\mathcal{K}\)-functional and its equivalent modulus of continuities. In Section
4, we investigate the uniform convergence of the operators and present the rate of convergence via the weighted modulus of continuities. In the last section, we give some pointwise estimates for the functions belonging to Lipschitz space.