#### Abstract

We give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke-type polynomials and obtain convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre's -functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.

#### 1. Introduction

The Szasz operators (also called Szasz-Mirakyan operators) which are defined by [1] where , , and have an important role in the approximation theory, and their approximation properties have been investigated by many researchers.

In [2], Jakimovski and Leviatan proposed a generalization of Szasz operators by means of the Appell polynomials which have the generating functions of the form: where is an analytic function in the disc , and . Under the assumption that for , Jakimovski and Leviatan [2], defined the following linear positive operators:

After that, Ismail [3] defined another generalization of Szasz operators involving the operators (1.1) and (1.3) by means of Sheffer polynomials. Let and be analytic functions in the disc . Here, and are real. The Sheffer polynomials are generated by With the help of these polynomials, Ismail constructed the following linear positive operators: under the assumptions(i)for , ,(ii) and .

Later, Varma et al. [4] defined another generalization of Szasz operators by means of the Brenke-type polynomials. Suppose that are analytic functions. The Brenke-type polynomials [5] have generating functions of the form from which the explicit form of is as follows:

Under the assumptions

Varma et al. introduced the linear positive operators via where and .

The aim of this paper is to present a Kantorovich type of the operators given by (1.10) and to give their some approximation properties. We consider the Kantorovich version of the operators (1.10) under the assumptions (1.9) as follows: where , , and . It is easy to see that defined by (1.11) is linear and positive.

In the case of and , with the help of (1.7), it follows that , so the operators (1.11) reduce to the Szasz-Mirakyan-Kantorovich operators defined by [6] Various approximation properties of the Szasz-Mirakyan-Kantorovich operators and their iterates may be found in [7–13].

The case of gives the Kantorovich version of the operators (1.3).

The structure of the paper is as follows. In Section 2, the convergence of the operators (1.11) is given by means of Korovkin's theorem. The order of approximation is obtained with the help of a classical approach, the second modulus of continuity, and Peetre's -functional in Section 3. Finally, as an example, we present a Kantorovich type of the operators including Gould-Hopper polynomials and then we give a Voronovskaya-type theorem for the operators including Gould-Hopper polynomials.

#### 2. Approximation Properties of Operators

In this section, we give our main theorem with the help of Korovkin theorem. We begin with the following lemma which is necessary to prove the main result.

Lemma 2.1. *For all , the operators defined by (1.11) verify
*

*Proof. *Using the generating function of the Brenke-type polynomials given by (1.7), we can write
From these equalities, the assertions of the lemma are obtained.

Lemma 2.2. *For , one has
*

*Proof. *From the linearity of , we get
Next, we apply Lemma 2.1.

Theorem 2.3. *Let
**
If , then
**
and the operators converge uniformly in each compact subset of . *

*Proof. *Using Lemma 2.1 and taking into account the equality (2.8) we get
The above convergence is satisfied uniformly in each compact subset of . We can then apply the universal Korovkin-type property (vi) of Theorem 4.1.4 in [14] to obtain the desired result.

#### 3. The Order of Approximation

In this section, we deal with the rates of convergence of the to by means of a classical approach, the second modulus of continuity, and Peetre's -functional.

Let . If , the modulus of continuity of is defined by where denotes the space of uniformly continuous functions on . It is also well known that, for any and each ,

The next result gives the rate of convergence of the sequence to by means of the modulus of continuity.

Theorem 3.1. *Let . The operators satisfy the following inequality:
**
where
*

*Proof. *Using (2.1), (3.2), and the linearity property of operators, we can write
By using the Cauchy-Schwarz inequality for integration, we get
which holds that
By applying the Cauchy-Schwarz inequality for summation on the right-hand side of (3.7), we have
where is given by (3.4). If we use this in (3.5), we obtain
On choosing , we arrive at the desired result.

Recall that the second modulus of continuity of is defined by where is the class of real valued functions defined on which are bounded and uniformly continuous with the norm .

Peetre's -functional of the function is defined by where and the norm (see [15]). It is clear that the following inequality: holds for all . The constant is independent of and .

Theorem 3.2. *Let . The following
**
holds, where
*

*Proof. *From the Taylor expansion of , the linearity of the operators and (2.1), we have
Since
for , by considering Lemmas 2.1 and 2.2 in (3.16), we can write that
which completes the proof.

Theorem 3.3. * Let . Then
**
where
**
and is a constant which is independent of the functions and . Also, is the same as in Theorem 3.2. *

*Proof. *Suppose that . From Theorem 3.2, we can write
The left-hand side of inequality (3.21) does not depend on the function , so
where is Peetre's -functional defined by (3.11). By the relation between Peetre's -functional and the second modulus of smoothness given by (3.13), inequality (3.21) becomes
whence we have the result.

*Remark 3.4. * Note that when , then , , and tend to zero in Theorems 3.1–3.3 under the assumption (2.8).

#### 4. Special Cases and Further Properties

Gould-Hopper polynomials [16], which are -orthogonal polynomial sets of Hermite type [17], are generated by from which it follows that where, as usual, denotes the integer part.

In [4], the authors showed that the Gould-Hopper polynomials are Brenke-type polynomials with and , and the restrictions (1.9) and condition (2.8) for the operators given by (1.10) are satisfied under the assumption . These operators including the Gould-Hopper polynomials are as follows: where .

The special case and of (1.11) gives the following Kantorovich version of including the Gould-Hopper polynomials: under the assumption .

*Remark 4.1. * For , we find and the operators given by (4.4) reduce to the Szasz-Mirakyan-Kantorovich operators given by (1.12).

Now, we give a Voronovskaya-type theorem for the operators (4.4). In order to prove this theorem, we need the following lemmas.

Lemma 4.2. *For the operators , one has
*

*Proof. *From the generating function (4.1) for the Gould-Hopper polynomials, one can easily find the above equalities.

Lemma 4.3. * For , one has
*

*Proof. *It is enough to use Lemma 4.2 to obtain above equalities.

Theorem 4.4. * Let . Then one has
*

*Proof. *By Taylor's theorem, we get
where and . If we apply the operator to the both sides of (4.8), we obtain
In view of Lemmas 4.2 and 4.3, the equality (4.9) can be written in the form
where
Applying Cauchy-Schwarz inequality, we get
If we use Cauchy-Schwarz inequality again on the right-hand side of the inequality above, then we conclude that
In view of Lemma 4.3,
holds. On the other hand, since and , then it follows from Theorem 2.3 that
Considering (4.13), (4.14), and (4.15), we immediately see that
Then, taking limit as in (4.10) and using (4.16), we have
which completes the proof.

*Remark 4.5. * Getting in Theorem 4.4 gives a Voronovskaya-type result for the Szasz-Mirakyan-Kantorovich operators given by (1.12).