ON A NEW APPROACH BY MODIFIED (p; q)-SZÁSZ-MIRAKYAN OPERATORS

In this paper, we introduce a new type of (p; q) exponential function with some properties and a modified (p; q)-Szász-Mirakyan operators by virtue of this function by investigating approximation properties. We obtain moments of generalized (p; q)-Szász-Mirakyan operators. Furthermore, we derive direct results, rate of convergence, weighted approximation result, statistical convergence and Voronovskaya type result of these operators with numerical examples. Graphical representations reveal that modified (p; q)-Szász-Mirakyan operators have a better approximation to continuous functions than pioneer one.


Introduction
Approximation theory is one of the oldest branches of mathematics. To approximate continuous functions with q-analogue of linear positive operators is significant application of q-calculus in approximation theory.
Cieśliński [1] established alternative definition of q-exponential function. He defined q-exponential function using Cayley transformation. The main advantages of the new q-exponential function consist of better qualitative properties i.e., its properties are more similar to properties of e z , z ∈ C [1]. Over the years, many research papers were developed on q-analogue of various linear positive operators and their approximation properties.
Recently, in [2] research of Bernstein-Stancu operators on (p; q)-integers were performed and discussed uniform convergence and direct result of the operators. Eventually, in [3] (p; q)-analogue of Bernstein operators was investigated and developed the same convergence. Acar [5] and Mursaleen et.al [4], [12] proposed (p; q)-generalization of Szász-Mirakyan operators and discussed uniform convergence, rate of convergence, Voronovskaya result in those papers.
The motivation of recent work is developing a new type of (p; q) exponential function and utilizing this new exponential function to modify (p; q)-Szász-Mirakyan operators. We studied uniform convergence and statistical convergence of modified (p; q)-Szász-Mirakyan operators. In the first section, we discussed some sequences and rate of convergence of operators. We also proved Voronovskaya type result. In the last section, we present some graphical representations.
Another way of defining two (p; q)-exponentials as infinite products is 2. New type of (p; q)-exponential function New (p; q)-exponential function is determined as e p,q (z), E p,q (z) are usual (p; q)-exponential functions. where, .
The above (p; q)-exponential function (2.1) has more improved properties similar to function e z .
The definition of (p, q)-Szász-Mirakyan operators in [5] is Acar obtained moments, uniform convergence and Voronovskaya result of the above operators.
We define a different sort of modified (p, q)-Szász-Mirakyan operators via new (p; q)-exponential function Remark 2.1. We choose an x between 0 and p n−1 +q n−1 p n −q n because we want E p,q ([n] p,q x) to be convergent.

Remark 2.2. From calculations for every k ∈ N;
[k]p,q Then we consider Clearly, s n (p, q; x) is positive for 0 < q < p ≤ 1, n ∈ N and every 0 ≤ x < 2 (p−q)[n]p,q . The operator S n,p,q is linear and positive.
3. Moments of S n,p,q Here, we determine approximation moments of operators (2.12).
Lemma 3.1. For n ∈ N and 0 < q < p ≤ 1. Below equalities are verified: Central moments are: Remark 3.1. From our choice of p and q, we know that lim n→∞ [n] p,q = 1 p−q . But, to get the uniform convergence and other results of approximation for S n,p,q we suppose that sequences q n ∈ (0, p n ); p n ∈ (q n , 1] such that q n , p n → 1 and p N n → a, q N n → b as n tending to infinity, i.e., lim n→∞ 1/[n] p,q = 0.

Now, we have uniform convergence of new kind of operators for all
Theorem 3.1. Let (p n ) and (q n ) be the sequences such that p n → 1, q n → 1 and p N n → a, q N n → b as n tending to infinity then for each f ∈ C ϑ [0, ∞) Proof. From Korovkin's result, we put evidence that lim n→∞ S n,p,q (t i ) − x i ϑ = 0, i = 0, 1, 2.
Since S n,p,q (1; x) = 1, the result is clear for i = 0.

Some consequences
For this section, we provide several results on local approximation for S n,p,q (f ; x). Here, Peetre's K-functional is given by is the second order modulus of continuity of functions f in C b [0, ∞). The first order modulus of continuity inequality is satisfied.
Proof. From the definition of S n,p,q (f ; x), Applying supremum to both sides here One has Theorem 4.2. Let (p n ) and (q n ) be the sequences such that p n → 1, q n → 1 and p N n → a, q N n → b as n tending to infinity. Then for f ∈ C b [0, ∞), there exists A > 0 such that Proof. For h ∈ W 2 , using Taylor's expansion Taking infimum of the right hand side of above inequality for all h ∈ W 2 , [n]p,q .

Rate of convergence
Suppose that C[0, ∞) is set of all continuous functions on [0, ∞) and consider following sets: The first order modulus of continuity on [0, a] is defined as . Let (p n ) and (q n ) be the sequences such that p n → 1, q n → 1 and p N n → a, q N n → b as n tending to infinity and ω a+1 (f, δ) be the modulus of continuity on [0, a + 1] ⊂ [0, ∞). Then Combining above inequality and Cauchy-Schwarz inequality,

From the central moments of operators and for
[n]p,q .

Weighted approximation result
Here, we discuss weighted approximation of S n,p,q through polynomial weight over the space C M defined below.

Statistical convergence
In this section, we obtain statistical convergence for new modified (p; q)-Szász-Mirakyan operators. We need the following theorem [10] Then Now, the result on statistical convergence of the operators defined in (2.12).
Theorem 7.2. Suppose that S n,p,q (f ; x) is defined as in (2.12), where (p n ) and (q n ) are the sequences such that p n → 1, q n → 1 and p N n → a, q N n → b as n tending to infinity. Then Proof. From above theorem, we only have to prove that From the moments of S n,p,q (f ; x), it is obvious that the result is true for j = 0, 1.
Then δ{k ≤ n : S n,p,q (t 2 ; x) − x 2 C b ≥ } ≤ δ{k ≤ n : But the right hand side of the above inequality is zero because st − lim The theorem is proved.

Voronovskaya type result
Theorem 8.1. Let (p n ) and (q n ) be the sequences such that p n → 1, q n → 1 and p N n → a, q N n → b as n tending to infinity then for each function f, f , f ∈ C * ϑ [0, ∞) is uniformly convergent on [0, a], a > 0.

Proof. Consider Taylor's formula on
where P (t, x) is Peano's remainder, P (t, x) → 0 as t → x.
Also, the absolute error can be seen graphically in the Figure 5 and 6.

Acknowledgements
The second author would like to express his gratitude to King Khalid University, Abha, Saudi Arabia, for providing administrative and technical support.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.