(p, q)-Generalization of Szász–Mirakjan operators and their approximation properties

*Correspondence: mustafa.kara@emu.edu.tr 1Department of Mathematics, Eastern Mediterranean University, Gazimagusa TRNC, Turkey Abstract We introduce a new modification of (p,q)-analogue of Szász–Mirakjan operators. Firstly, we give a recurrence relation for the moments of (p,q)-analogue of Szász–Mirakjan operators and present some explicit formulae for the moments and central moments up to order 4. Next, we obtain quantitative estimates for the convergence in the polynomial weighted spaces. In addition, we give the Voronovskaya theorem for the new (p,q)-Szász–Mirakjan operators.


Introduction
In the last two decades, quantum calculus plays a significant role in the approximation of functions by a positive linear operator. In 1987, Lupaş [1] introduced the Bernstein (rational) polynomials based on the q-integers. In 1996, Phillips [2] introduced another generalization of Bernstein polynomials based on q-integers. In , in the case of 0 < q < 1, many operators have been introduced and examined. Among the most important operators there are q-Szász operators. In [18][19][20][21] the authors constructed and studied different q-generalizations of Szász-Mirakjan operators in the case 0 < q < 1. In 2012, Mahmudov [24] introduced the q-Szász operator in the case q > 1 and studied quantitative estimates of convergence in polynomial weighted spaces and gave the Voronovskaya theorem.

Construction of K l,p,q and moment estimations
We give some basic notations and definitions of the (p, q)-calculus.
The (p, q)-integer and (p, q)-factorial are defined by The (p, q)-derivative D p,q g of a function g(z) is defined by The product and quotient formulae for the (p, q)-derivative are as follows: It is known that The (p, q)-analogues of an exponential function, denoted by e p,q (z) and E p,q (z), are defined by and the (p, q)-derivatives of e p,q (az) and E p,q (z) are D p,q e p,q (az) = ae p,q (apz), D p,q E p,q (az) = ae p,q (aqz).
For any integer l, and D p,q (zy) 0 p,q = 0. The formula of the kth (p, q)-derivative of the polynomial (zy) l p,q is where l ∈ Z + and 0 ≤ k ≤ l. The (p, q)-analogue of the Taylor formulas for any function g(z) is defined by Let C β denote the set of all real-valued continuous functions g on [0, ∞) such that w β g is bounded and uniformly continuous on [0, ∞) endowed with the norm where w 0 (z) = 1, and w β (z) = 1 1+z β for β ∈ N. The corresponding Lipschitz class is given for 0 < α ≤ 2 by 2 j g(z) := g(z + 2j) -2g(z + j) + g(z), Now we introduce the (p, q)-Szász-Mirakjan operator.
It is clear that the operator K l,p,q is linear and positive. It is known that the moments K l,p,q (t m ; z) play a fundamental role in the approximation theory of positive operators.
Lemma 2 Let 0 < q < p ≤ 1 and m ∈ N. We have the following recurrence formula: Proof According to the definition of K l,p,q (8), we have Next, we use the identity q[k] p,q + p k = [k + 1] p,q to get the desired formula , l ∈ l, and k ≥ 0. We have the following identities related to the (p, q)-derivative: Proof We take the (p, q)-derivative of s lk (p, q; z): Then Using the obtained formula and the definition of the operator K l,p,q , we get the second desired formula: = p -l [l] p,q K l,p,q t m+1 ; pz -[l] p,q p -(l-1) zK l,p,q t m ; pz .
For i = 2, 3, 4, recurrence formula (10) gives us the following results: p,q p -2(l-1) , K l,p,q t 4 ; z = z p -(l-1) [l] p,q D p,q K l,p,q t 3 ; z p + zK l,p,q t 3 ; z Proof In fact, we may easily calculate third-and fourth-order central moments as follows: K l,p,q (tz) 3 ; z = K l,p,q t 3 ; z -3zK l,p,q t 2 ; z + 3z 2 K l,p,q (t; z)z 3 In our study, we assume that q = q l ∈ (0, 1) and p = p l ∈ (q, 1] are such that For all 0 < q < p ≤ 1 and j ≥ 0, the (p, q)-difference operators are defined as 0 p,q g(z j ) = 0, 1 p,q g(z j ) = p,q g(z j ), and k+1 p,q g(z j ) = p k k p,q g(z j+1 )q k k p,q g(z j+1 ), [l] p,q . Using this definition, we can prove the following lemmas.

Lemma 10
The (p, q)-Szász-Mirakjan operator can be represented as Proof Indeed, The next result gives an explicit formula for the moments K l,p,q (t m ; z) in terms of Stirling numbers, which is a (p, q)-analogue of Becker's formula; see [46].

Lemma 11
For 0 < q < p ≤ 1 and m ∈ l, we have where are the second-type Stirling polynomials satisfying the equalities Clearly, K l,p,q (t m ; z) are polynomials of degree m without a constant term.
Using mathematical induction, assume (13) to be valued for m. Then from Lemma 3 we get Remark 12 For p = q = 1, formulae (14) become recurrence formulas satisfied by the second-type Stirling numbers from [8].
Thus K l,p,q is a linear positive operator from C β into C β .

Convergence of (p, q)-Szász-Mirakjan operators
In [47,Theorem 1] and [48,Theorem1], Totik and de la Cal investigated the class problem of all continuous functions g such that K l,p,q (g) converges to g uniformly on the whole interval [0, ∞) as l → ∞. The following thorem is a (p, q)-analogue of Theorem 1 in [48].
Therefore K l,p,q (g; z) converges to g uniformly on [0, ∞) as l → ∞ whenever g * is uniformly continuous.
The proof is completed.