#### Abstract

The present study introduces generalized -Bernstein–Stancu-type operators with shifted knots. A Korovkin-type approximation theorem is given, and the rate of convergence of these types of operators is obtained for Lipschitz-type functions. Then, a Voronovskaja-type theorem was given for the asymptotic behavior for these operators. Finally, numerical examples and their graphs were given to demonstrate the convergence of to with respect to values.

#### 1. Introduction

The Bernstein operators, which are positive linear operators, are of great importance for the theory of approximation. In [1], Bernstein operators were introduced by Bernstein to prove the Weierstrass approximation theorem. For , the classical Bernstein operators are given aswhere the Bernstein basis functions are defined as

There are many generalizations of the operators. In [2], Gadjiev and Ghorbanalizadeh gave Bernstein–Stancu operators with shifted knots:where and for , and are positive real numbers, and for . In the case of for , operator (3) reduces to the classical Bernstein operator (1). In addition, the case of was handled and examined by Stancu in [3] and these operators were called the classical Bernstein–Stancu operators. A Durrmeyer variant of the Bernstein–Stancu operators in (3) was studied by Dinlemez Kantar and Ergelen in [4], and a Voronovskaja-type approximation theorem for these operators was given. Several studies were conducted on some approximation properties, and asymptotic-type results were given for these operators in [5–14].

In [5], Cai et al. introduced the Bernstein operators with shape parameter as follows:where and are Bézier basis functions with shape parameter defined by

Later in [6], Cai proposed Kantorovich-type -Bernstein operators, as well as their Bézier variant, and examined the approximation results. Kantorovich-type -Bernstein operators were also studied by Acu et al. in [7], and they considered the approximation properties and asymptotic-type results.

Recently, in [8], Srivastava et al. constructed -Bernstein–Stancu operators defined bywhere and Bézier basis functions with shape parameter are defined in (5).

In the present study, we introduce the following generalized -Bernstein–Stancu operators with shifted knots for :where and are positive real numbers satisfying for and Bézier basis functions with shape parameter are defined bywhere for , are positive real numbers, for , and . We give approximation properties and Voronovskaja-type approximation theorem for asymptotic behavior of operator (7). When for and , operator (7) reduces to the classical Bernstein operator (1). When , it reduces to the Bernstein–Stancu operator (3). When for , it reduces to the Bernstein operator with shape parameter (4). In the following section, several lemmas are given to prove the main results.

#### 2. Some Preliminary Results

Lemma 1. *For generalized -Bernstein–Stancu operators with shifted knots, we have the following equalities:*

*Proof. *If we use Bézier basis functions (8) in -Bernstein–Stancu operators (7), we obtainwhereThanks to the linearity of Bernstein–Stancu operators (3), we obtainNow, we will compute and Using (12), , and in , we obtain the following equation:Then, we have the following equality for the third moment by using the linearity of :whereFrom the linearity of the Bernstein–Stancu operators (3)and Bernstein basis functions, we obtainNext, we compute and Combining (17), , and , we obtain the result for

Using the same technique in the above moments, we obtain and

Corollary 1. *Using Lemma 1, we obtain the following inequalities of central moments for and for fixed and :**Using Lemma 1 and the linearity of , we have the following Corollary 2.*

Corollary 2. *We obtain the following equalities:* * ** *

#### 3. Convergence Properties of

For the asymptotic behavior of operators, we give the following Korovkin-type approximation theorem.

Theorem 1. *If and , then operators converge uniformly to on , where is a Banach space of all continuous functions on with norm .*

*Proof. *Using the equalities , and of Lemma 1, we obtainTherefore, the proof is completed using Korovkin theorem.

For , the Pee