A Kantorovich-Stancu Type Generalization of Szasz Operators including Brenke Type Polynomials

have an important role in the approximation theory [1]. Their Korovkin type approximation properties and rates of convergence have been investigated by many researchers. Recently, there is a growing interest in defining linear positive operators via special functions (see [2–13]). In particular, many authors have studied various generalizations of Szasz operators via special functions. In [14], Jakimovski and Leviatan constructed a generalization of Szasz operators by means of the Appell polynomials.Then, Ismail [15] presented another generalization of Szasz operators by means of Sheffer polynomials, which involves the operators (1) defined by Jakimovski and Leviatan in [14]. In [11],Varma et al. considered the following generalization of Szasz operators by means of the Brenke type polynomials, which are motivated by the operators defined by Jakimovski and Leviatanand Ismail, for x ≥ 0 and n ∈ N:


Introduction
For each positive  and  ∈   ([0, ∞)) or ([0, ∞)) ∩ , the Szasz-Mirakyan operators defined by have an important role in the approximation theory [1].Their Korovkin type approximation properties and rates of convergence have been investigated by many researchers.
Recently, in [8], the Kantorovich type of the operators given by (2) under the assumptions (3) has been defined as (8) where  ∈ N,  ≥ 0 and  ∈ [0, ∞), and some of its properties have been investigated.
The purpose of this study is to introduce a Kantorovich-Stancu type modification of the operators given by ( 8) and to examine the approximation properties of these operators.We also present a Kantorovich-Stancu type of the operators including Gould-Hopper polynomials and then we prove a Voronovskaya type theorem for these operators including Gould-Hopper polynomials.
In the case of  =  = 0, the operator (10) turns out to be the Szasz-Mirakyan-Kantorovich operators given by (7).

𝑛
given by ( 9), we give some results which are necessary to prove the main theorem.Lemma 1. Kantorovich-Stancu type operators, defined by (9), are linear and positive.Lemma 2. For each  ∈ [0, ∞), the Kantorovich-Stancu type operators (9) have the following properties: By using these equalities, we obtain the assertions of the lemma by simple calculation.
Proof.According to Lemma 2, by considering the equality ( 16), we get lim This convergence is satisfied uniformly in each compact subset of [0, ∞).Then, the proof follows from the universal Korovkin-type property (vi) of Theorem 4.1.4in [28].

Rates of Convergence
In this section, we compute the rates of convergence of the operators  (,)  () to  by means of a classical approach, the second modulus of continuity, and Peetre's -functional.
Let  ∈ C[0, ∞).Then for  > 0, the modulus of continuity of  denoted by (; ) is defined to be where C[0, ∞) denotes the space of uniformly continuous functions on [0, ∞).Then, for any  > 0 and each  ∈ [0, ∞), it is well known that one can write The next result gives the rate of convergence of the sequence  (,)  () to  by means of the modulus of continuity. where Proof.Using linearity of the operators  (,)  , ( 11) and ( 20  According to the Cauchy-Schwarz inequality for integration, we obtain that from which, it follows that By using the Cauchy-Schwarz inequality for summation on the right hand side of ( 25  (( − ) 2 ; )) where   () is given by (22).Considering this inequality in (23), we find that If we set  = √  (), the proof is completed.
Now, we will study the rates of convergence of the operators  (,)  to  by means of the second modulus of continuity and Peetre's -functional.
Recall that the second modulus of continuity of  ∈   [0, ∞) is defined by where   [0, ∞) is the class of real valued functions defined on [0, ∞) which are bounded and uniformly continuous with the norm where  and satisfy the generating function where, as usual, [⋅] denotes the integer part [30].The Gould-Hopper polynomials are Brenke-type polynomials for the special case of () =  ℎ +1 and () =   in (5).From (2), the operators including the Gould-Hopper polynomials are as follows: where  ∈ [0, ∞) and ℎ ≥ 0 (see [11]).