#### Abstract

In the present paper, the generalized -gamma-type operators based on -calculus are introduced. The moments and central moments are obtained, and some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre -functional. Also, the rate of convergence, weighted approximation, and pointwise estimates of these operators are studied. Finally, a Voronovskaja-type theorem is presented.

#### 1. Introduction

In [1], Mazhar introduced gamma operators preserving linear functions as follows: where , , . In [2], Karsli considered new gamma operators preserving as follows: where . In [3], Mao defined generalized gamma operators as follows: where . Obviously, and . In [4, 5], some approximation properties of operators ((1)â€“(3)) were discussed.

In Bernstein polynomials, some of their modifications and corresponding operators have been studied in many papers (see [6â€“10]). The -analogue of well-known positive probability operators were widely studied and discussed (see books [11â€“13]) since Bernstein polynomials were proposed by Lupa [14] and Phillips [15]. In [16], the -analogue of the operators (1) was defined and discussed. In [17], Cai and Zeng constructed and studied a -analogue of the operators (2). Meantime, modifications and generalizations of the operators (2) were introduced and researched in [18â€“20]. In [21], Karsli constructed a -analogue of the operators (3) and extended the works of [16, 17, 20]. Recently, many operators are constructed with two-parameter -integers based on postquantum calculus (-calculus) which have been used widely in many areas of sciences such as Lie group, different equations, hypergeometric series, and physical sciences. First, we recall some useful concepts and notations from -calculus, which can be found in [11â€“13]. The -integers are defined by

By some simple calculation, for any , we have the following relation:

The -factorial is defined by

The -power basis is defined by

Let be an arbitrary function. The improper -integral of on is defined as (see [22])

Let be a nonnegative integer. The -gamma function is defined as

Aral and Gupta [23] proposed a -beta function of the second kind for , as and gave the relation of the -analogues of beta and gamma functions:

As a special case, if , . It is obvious that the order is important for the -setting, which is the reason why the -variant of beta function does not satisfy commutativity property, i.e., .

Since Mursaleen et al. firstly introduced -calculus in approximation theory and constructed the -analogue of Bernstein operators [24] and -Bernstein-Stancu operators [25], generalizations of many well-known approximation operators based on -calculus were widely introduced and discussed by several authors (see -SzÃ¡sz-Mirakjan operators [26], -Baskakov-Durrmeyer-Stancu operators [27], -Bernstein-Stancu-Schurer-Kantorovich operators [28], -Baskakov-beta operators [29], -Lorentz polynomials [30], -SzÃ¡sz-Mirakjan Kantorovich operators [31], -Bleimann-Butzer-Hahn operators [32], -Bernstein operators [33, 34], and so on). In [35], Cheng and Zhang constructed a -analogue of the operators (1) using the -beta function of the second kind and studied their approximation properties. Later, Cheng et al. defined the -analogue of the operators (2) and researched their approximation properties in [36]. All these achievements motivate us to construct the -analogue of the gamma operator (3) and generalize the works of [35, 36]. Now, we construct generalized -gamma-type operators as follows:

*Definition 1. **Let*, , , and *. For*, then the -analogue of the gamma operator (3) can be defined by
In the case , we obtain the operators (1); in the case , we obtain the operators (2); in the case , we obtain the operators (3); in the case , we obtain the operators [16]; in the case , we obtain the operators [17]; in the case , we obtain the operators [21]; in the case , we obtain the operators [35]; and in the case , we obtain the operators [36].

The paper is organized as follows: In Section 1, we introduce the history of gamma-type operators and recall some basic notations about -calculus; then, we construct the generalized -gamma operators with the -beta function. In Section 2, we obtain the auxiliary lemmas and corollaries about the moment computation formulas. And the second- and fourth-order central moment computation formula and limit equalities are also obtained. In Section 3, we discuss the local approximation about the operators by means of modulus of continuity and Peetre -functional. In Sections 4 and 5, the rate of convergence and weighted approximation for these operators are researched. In Section 6, two pointwise estimates are given by using the Lipschitz-type maximal function. In Section 7, the Voronovskaja-type asymptotic formula is presented.

#### 2. Moment Estimates

In order to obtain the approximation properties of the operators , we need the following lemmas and corollaries.

Lemma 2. *For , , , , and , we have
*

*Proof. *Set , , and , we have

Then, the following corollary can be obtained immediately.

Corollary 3. *For , , and , the following equalities hold:
*

Corollary 4. *For , , and , using Corollary 3, we can easily obtain the following explicit formulas for the first and second central moments:
*

Corollary 5. *The sequences satisfy such that and as ; then, for any , we have
*

*Proof. *Using (5), we have . Hence,
Using and , we have
Thus,
Now, we prove the limit equality (19) while is similar. Set and , by (5); we can obtain
Hence, we can rewrite
Further, we can easily get
For any , . We can obtain
we obtain the required result.

Corollary 6. *Let us denote the norm on (the class of real valued continuous bounded functions on ). For any , we have
*

*Proof. *In view of (12) and Corollary 3, the proof of this corollary can be obtained easily.

#### 3. Local Approximation

For any , let us consider the following -functional: where and . The usual modulus of continuity and the second-order modulus of smoothness of can be defined by

By [37] (p.177, Theorem 2.4), there exists an absolute constant such that

Theorem 7. *Let be the sequences defined in Corollary 5 and . Then, for all , there exists an absolute positive such that
*

*Proof. *Define the following new operators:
Let and . By Taylorâ€™s expansion formula, we get
Applying to the above equality and , we can obtain
By Corollary 6 and (32), we easily know . Hence,
Taking the infimum on the right-hand side over all and using (30), we obtain the desired assertion.

Corollary 8. *Let be the sequences defined in Corollary 5 and . Then, for any finite interval , the sequence converges to uniformly on .*

#### 4. Rate of Convergence

Let where is the weighted function given by and is an absolute constant depending only on . is equipped with the norm . As is known, if is not uniform, we cannot obtain . In [38], Ispir defined the following weighted modulus of continuity: and proved the properties of monotone increasing about as , , and the inequality while and . Meantime, we recall the modulus of continuity of on the interval by

Theorem 9. *Let , , and , we have
*

*Proof. *For any and , we easily have ; thus,
and for any , , and , we have
For (41) and (42), we can get
By Schwarzâ€™s inequality, for any , we can get
By taking and the supremum over all , we accomplish the proof of Theorem 9.

#### 5. Weighted Approximation

In this section, we will discuss the following three theorems about weighted approximation for the operators :

Theorem 10. *Let and the sequences satisfy such that as ; then, there exists such that for all and , the inequality
holds.*

*Proof. *Using (37) and (38), we can write
For any and , (46) can be rewritten:
Using (18) and (19), there exists such that for any ,