Abstract

In the present article, we construct -Szász-Mirakjan-Kantorovich-Stancu operators with three parameters . First, the moments and central moments are estimated. Then, local approximation properties of these operators are established via -functionals and Steklov mean in means of modulus of continuity. Also, a Voronovskaja-type theorem is presented. Finally, the pointwise estimates, rate of convergence, and weighted approximation of these operators are studied.

1. Introduction

During this decades, the applications of -calculus transpired as a new area in the field of operator approximation theory. Many researchers constructed and discussed many positive linear operators based on -integers, -exponential functions, -Gamma functions [1], -Beta functions, and so on. Since Mursaleen et al. first constructed -Bernstein operators [2] and -Bernstein-Stancu operators [3], several generalizations of well-known positive linear operators based on -calculus have been introduced and studied (see [4–11]). In [12], Acar first proposed -Szász-Mirakjan operators defined on . In [13], Kara et al. constructed a modified -Szász-Mirakjan as follows: where , and . Certain basic notations of -calculus are mentioned below (for details see [14]): For each real number , -analogue of named is defined by

And for each nonnegative integer , the -integer and -factorial are defined by

The -analogue of the exponential function is defined by

Let be an arbitrary function and . The -Jackson integral [15] was defined by

And the -Jackson integral over an interval can be defined by

We easily know that -Jackson integral (6) is not positive unless it is assumed that is a nondecreasing function. To solve this problem, Acar et al. [16] defined the -integral of the arbitrary function on interval as follows:

It is obvious that integral (6) and integral (7) of on are equivalence.

The Kantorovich modification of positive linear operators on is a method to approximate the Riemann integrable functions. The idea behind the Kantorovich modifications mainly depends on replacing the sample value by (see [17, 18]). By definite integral substitution, we have . However, two Kantorovich modifications may be not equivalence or cannot use definite integral substitution in -calculus and -calculus. For the researches about -Szász-Mirakjan-Kantorovich-operators, we can see [19–21]. Meantime, the idea behind the Stancu modifications mainly depends on replacing the sample value by with two parameters (see [22]). For the researches about the Stancu modification of -operators, we can see [23, 24]. All these achievements motivate us to construct the Stancu and Kantorovich generalizations of -Szász-Mirakjan (1) with three parameters as follows:

Definition 1. For , , , and , the -Szász-Mirakjan-Kantorovich-Stancu operators can be defined by

2. Auxiliary Results

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

Lemma 2. For , , , we have .

Proof. Using (7),

Lemma 3. ([13], Lemma 4) For , , and , we have

The following lemma will tell us the relation between the moment of the operators and the moment of the operators :

Lemma 4. For , , , , , we have the following recursive relation:

Proof. By direct computation, we have Hence, the proof of Lemma 4 is completed.

Then, the following lemma can be obtain immediately:

Lemma 5. For, , , , we have

Lemma 6. Under the condition of Lemma 5, we can easily obtain the following formulas for the first and second central moments:

Lemma 7. The sequences , satisfy , such that , , as ; then for any , , , , we have

Proof. By , we have . Thus, we easily obtain (15) and (16). As , we can rewrite Set . Applying Lemma 4 and , , we can also rewrite Combining , we can obtain we obtain the required result.