In this paper, we study the Kantorovich-Stancu-type generalization of Szász-Mirakyan operators including Brenke-type polynomials and prove a Korovkin-type theorem via the -statistical convergence and power series summability method. Moreover, we determine the rate of the convergence. Furthermore, we establish the Voronovskaya- and Grüss-Voronovskaya-type theorems for -statistical convergence.

1. Introduction and Preliminaries

Let (set of natural numbers) and . Then, the natural density or we can say asymptotic density of is defined by whenever the limit exists, where denotes the cardinality of the set . A sequence is statistically convergent to if for every and we write .

Let be a matrix and be a sequence. The transform of the sequence is defined by if the series converges for every . We say that is summable to the number if converges to . The summability matrix is regular whenever .

Let be a regular matrix. A sequence is said to be -statistically convergent (see [1]) to real number if for any , and we write . If is Cesàro matrix of order 1, then -statistical convergence is reduced to the statistical convergence.

In this paper, we also use the power series summability method which includes several known summability methods such as Abel and Borel (see [29]). Note that the power method is more effective than the ordinary convergence (see [10]).

Let be a sequence of real numbers such that , , and the corresponding power series has radius of convergence with . If for all , then we say that is convergent in the sense of power series method (see [11, 12]). Define , where and are analytical functions such that and for all (see [13]). Clearly, . Moreover, the power series method is regular if and only if holds for each (see [14]).

We study a Korovkin-type theorem for the Kantorovich-Stancu-type Szász-Mirakyan operators via power series method. We determine the rate of convergence for these operators. Furthermore, we give a Voronovskaya-type theorem for statistical convergence. Such type of operators is widely studied by several authors (see [1519]).

We start by recalling the class of Kantorovich-Stancu-type generalization of Szász-Mirakyan operators, including Brenke-type polynomials. For every , for and , where . In what follows, we calculate the moments and central moments for Kantorovich-Stancu of Szász-Mirakyan operators. Let us mention some properties of the functions and (see [13, 20]). (1), , for all and (2)(3)Series for and for are convergent for and are analytic functions

The next lemma is followed immediately from the fact that .

Lemma 1. Let be the operator . For all , For example, Lemma 1 for gives

Theorem 2. Let for all and let be the operator . Then,

Proof. By the definition of the operators, we have Thus, by Lemma 1, we complete the proof.

Lemma 3 (for instance, see [21], equation (1.27)). Let be two operators on the set of functions defined by and . Then, Moreover, where is the Stirling number of the second kind.
Define and , for all . Therefore, Theorem 2 with and Lemma 3 imply the following theorem.

Theorem 4. Let for all . Then, where is the Stirling number of the second kind.

Example 5. By applying Theorem 4 for with using (2), we obtain

Theorem 6. Let , and let be the operator . Then,

Proof. By the definitions, we have Thus, Lemma 1 completes the proof.
By Theorem 6 and Lemma 3, we obtain the following result.

Theorem 7. Let . Then, where is the Stirling number of the second kind.

Remark 8. By applying Theorem 7 for , we obtain Theorem 6 for (with the help of mathematical programming), we obtain the following result.

Proposition 9. Let us consider that where is the -th derivative of Then, we obtain

Example 10. Let , , , , and . By the fact that , we have that .
Table 1 presents the values of the functions and at and , , where we approximated as We note that the Korovkin-type theorems are very useful tools in approximation which were studied in several function spaces [38, 10, 2229]. We say that sequence of operators converges to in the sense of power series if for every .

2. Main Results

We study here statistical convergence of the operators . Note that the Korovkin-type theorem for statistical convergence was considered in [24] as follows:

Theorem 11. Let be a sequence of positive linear operators on and let be a nonnegative regular summability matrix such that Then, for any where .
Based on the above theorem, we give the following result.

Theorem 12. Let be a regular matrix and be as in (2) on such that where denotes derivative and Then, for any where .

Proof. From Lemma 5, we have that . Now, we will estimate the following expressions: Note that . So from the last two relations we have that . Moreover, Now proof follows directly from Theorem 11.
This theorem is an extension of some known results for the Kantorovich-Stancu-type Szász-Mirakyan operators.

Example 13 (see [6]). Under conditions given in Theorem 12, we define the following operators where the sequence is given as follows: then By Theorem 11 we obtain , but the operators do not satisfy Theorem 12. Hence, the sequence is not statistically convergent but it is statistically convergent.

Remark 14. The sequence is not statistically convergent and hence not convergent. As an example, consider the Cesáro matrix of order 2. where This proves that is statistically convergent. We have By Example 13, this shows that does not satisfy Theorem 12.
In [27, 29], Korovkin-type theorems are proved by Abel summability method. Now, we discuss for power series method. Let () be the space of all bounded (continuous) functions on the interval .

Theorem 15. Let be a sequence of positive linear operators from into such that Then, for ,

Proof. Clearly, from (32) follows (31). Now, we show the converse that (31) implies (32). Let , then there exists a constant such that for all . Therefore, For every given , there exists such that whenever for all Define . If then From (33)–(35), we have that , namely, By applying the operator , is a monotone and linear operator, we obtain which implies On the other hand, From (38) and (39) we get Now, we estimate the following expression: By (40), we obtain Therefore, From the above relations and the linearity of , we obtain Hence, (32) follows from the last relation and (31).

3. Rate of Convergence

Modulus of continuity is defined by

It is not hard to verify

So, we can state the following.

Theorem 16. Let be a nonnegative regular summability matrix and . If is a sequence of positive real numbers such that , then where for any positive integer

Proof. Let By positivity and linearity of and (46), we see By applying the Cauchy-Schwartz inequality, we have Based on Examples 5 and Remark 8, we obtain