ON THE (p, q)−STANCU GENERALIZATION OF A GENUINE BASKAKOV- DURRMEYER TYPE OPERATORS

In this paper, we introduce a Stancu generalization of a genuine BaskakovDurrmeyer type operators via (p, q)− integer. We investigate approximation properties of these operators. Furthermore, we study on the linear positive operators in a weighted space of functions and obtain the rate of these convergence using weighted modulus of continuity.


Introduction
In the field of approximation theory, the quantum calculus has been studied for a long time.The generalization of (p, q)− calculus was introduced by Sahai and Yadav in [15].Recently, a series of papers giving (p, q)− generalizations a sequence of linear positive operators have been published in [3,4,[9][10][11][12][13].Our aim is to give Stancu type generalization, via (p, q)− integer, defined by Agrawal and Thamer as follows where in [5].
We refer reader to [2] for unexplained terminologies and notations.

Preliminaries and notations
Let's give a table of some basic formulas, motivated from q−calculus, used in (p, q)−calculus as the following Table1 (p, q)−calculus Relation with q−calculus [n] p,q = p n −q n p−q Recall that the beta function, introduced [14], in q− calculus is defined by where In the formula (2.2), K(A, n) = q n(n−1)/2 and K(A, 0) = 1 for n ∈ N. Inspiring the formula (2.1), we introduce (p, q)-beta functions B p,q (n, k), as a generalization of B q (n, k), A > 0 and n, k ∈ N\ {0}, defined by 3), then the formula is reduced to (2.1).
To obtain our main results, we need calculating the values of Korovkin monomial functions.
Lemma 3.1.The following equalities are satisfied for e m (t) = t m , m = 0, 1, 2 and n > 3 Proof.By the definition (p, q)−beta functions in (2.3), we obtain the estimate, If we apply the operators in (3.1) to the equality (3.2) for m = 0, we get And the proof of (i) is finished.With the direct computation, we obtain (ii) as follows: For (iii), Using the equality And so we have completed the proof of (iii).Now we consider that B[0, ∞) denotes the set of all bounded functions from [0, ∞) to R, B[0, ∞) is a normed space with the norm f B = sup{|f (x)| : x ∈ [0, ∞)} and C B [0, ∞) denotes the subspace of all continuous functions in B[0, ∞).We denote first modulus of continuity on finite interval The Peetre's K−functional is defined by The weighted Korovkin-type theorems were proved by Gadzhiev [8].We give the Gadzhiev's results in weighted spaces.
is a normed space with the norm exists finitely.
we get n ([n] pn ,qn +β) x(x + 1).And the proof of the Lemma 3.3 is now finished.
Thus we are ready to give direct results.Lemma 3.4.We have the inequality for every where δ α,β pn,qn,n (x) := n ([n] pn ,qn +β) x(x + 1).Proof.Using Taylor's expansion and the Lemma 3.2, we have the following equality On the other hand, combining the inequality and Lemma 3.3, we get n ([n] pn ,qn +β) x(x + 1), as desired.
Proof.For any g ∈ W 2 ∞ , we obtain the inequality Then, Lemma 3.4, we have Taking infimum over g ∈ W 2 ∞ on the right side of the above inequality and using the inequality (3.5), we reach the desired result.We have Then we get lim