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)–(3)) were discussed.

In Bernstein polynomials, some of their modifications and corresponding operators have been studied in many papers (see [6–10]). The -analogue of well-known positive probability operators were widely studied and discussed (see books [11–13]) 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 [18–20]. 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 [11–13]. 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 -Szász-Mirakjan operators [26], -Baskakov-Durrmeyer-Stancu operators [27], -Bernstein-Stancu-Schurer-Kantorovich operators [28], -Baskakov-beta operators [29], -Lorentz polynomials [30], -Szász-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.