Approximation by (p,q) Szász-beta–Stancu operators

Motivated by recent investigations, in this paper we introduce ( p ,  q )-Szász-beta–Stancu operators and investigate their local approximation properties in terms of modulus of continuity. We also obtain a weighted approximation and Voronovskaya-type asymptotic formula.

The ( p, q)-binomial coefficient is given by Definition 1. 1 The ( p, q)-power basis is defined below: Definition 1.2 [23] For d ≥ 0, the ( p, q)-gamma function is given as

Definition 1.3
The ( p, q) derivative of the function f is defined as , provided that f is differentiable at zero.

Proposition 1.4 [23] The ( p, q)-integration by parts is given by
The ( p, q)-beta function of second kind [4] is given by The relation between ( p, q)-beta and ( p, q)-gamma functions is given as .
For 0 ≤ x < ∞, 0 < q < p ≤ 1, Aral and Gupta [3] defined the ( p, q)-analogue of Szász-beta operators as follows: where s p,q The aim of this paper is to generalize the operator in (1) using Stancu-type parameters, (i.e. assuming 0 ≤ α ≤ β), we define: where s p,q n,k (x) is given in (1). In particular case, if α = β = 0, then the operators D p,q n,α,β ( f ; x) turn out to be the one defined by (1).

Moments
Lemma 2.1 [3] For the operator defined in (1), x ≥ 0 and for α = β = 0, the following equalities hold for , then for the operator in (2), we have the following moments.
Proof By the definition of operator (2) and Lemma 2.1, we have This completes the proof.
Using Lemma 2.2, we can obtain

Local approximation
Let us consider the space of all real valued continuous and bounded functions on R + and denote this space by C B (R + ) under the norm: Then, Peetre's K-functional is defined as Then as in [7], there exists a positive constant C such that The second-order modulus of smoothness of f ∈ C B (R + ) is and the usual modulus of continuity is given by where the auxiliary operators are given by and Proof By the definition of auxiliary operators, it can be shown that Let s ∈ W 2 . Then from the Taylor's expansion, we have Operating (8) with (5) and using (7), we get , q, x)). Therefore, Hence the proof is completed.
Now taking infimum on right-hand side over all s ∈ W 2 and using (4), we get Hence the proof of the theorem.
Proof Let f ∈ C[0, ∞) and 0 ≤ x < ∞. Then using monotonicity of the operator defined in (2), we can easily obtain for every δ > 0 that which is obtained using Lemma 2.2 and choosing δ n,β = 1 [n] p,q +β . Hence we arrive at the result.
If we put α = β = 0, we can find the similar results for the operators defined by (1): where δ n = 1 [n] p,q and it is observed that δ n,β ≤ δ n . Therefore, rate of convergence of D p,q n,α,β is better than D p,q n .

Weighted approximation
Let us consider the functions in weighted space defined as (2) C(R + ) be the set of all continuous functions f defined on [0, ∞).
is a normed vector space under the norm: Theorem 4.1 Let p = p n and q = q n such that 0 < q n < p n ≤ 1 and p n → 1, q n → 1 , p n n → 1, q n n → 1 as n → ∞. Then for each f ∈ C * 2 (R + ), we have  Hence (11) holds for λ = 2 Hence the theorem .