Szász–Durrmeyer Operators Involving Confluent Appell Polynomials

: This article is concerned with the Durrmeyer-type generalization of Szász operators, including confluent Appell polynomials and their approximation properties. Also, the rate of convergence of the confluent Durrmeyer operators is found by using the modulus of continuity and Peetre’s K -functional. Then, we show that, under special choices of A ( t ) , the newly constructed operators reduce confluent Hermite polynomials and confluent Bernoulli polynomials, respectively. Finally, we present a comparison of newly constructed operators with the Durrmeyer-type Szász operators graphically.


Introduction
As a polynomial set, an Appell set [1] satisfies the following criteria: the determining function that enables us to have is an official power series as follows: For some r > 0, it is presumed that the series in (1) convergent in |t| < r.Another way to describe the Appell polynomials is the P where {θ k } is unrelated to n with θ 0 ̸ = 0, an analytic function, A(t) has an extension of power series and is an analytical function, and is a confluent hypergeometric function.For all finite z, this function converges, assuming b / ∈ {. . . ,−1, 0} [2].Then, (a) k = a(a + 1) . . .(a + k − 1); k ≥ 1 (a) 0 = 1; 1 gives the definition of the Pocchammer symbol [2].

Approximation Properties
In this section, we give moments and central moments for our operator including confluent Appell polynomials.

Lemma 1. For any
Proof.One way to illustrate the proof is to use it as given in (2) By using these equalities in the operator, we obtain the desired results.
So, from the well-known Korovkin theorem [13] the proof is completed.
By using these equalities, the proof is completed.

Rate of Convergence
In this section, we give the rate of convergence by the modulus of continuity, Peetre's-K functional, and the second modulus of continuity, respectively.The modulus of continuity is given by It is due to the following feature of the modulus of continuity where Proof.Using the operators S η 's linearity and from Lemma 2, we obtain For integral by using the Cauchy-Schwarz inequality, it follows that Examining Cauchy-Schwarz disparity in summation, one can easily obtain where δ η (x) is given by (5).
can be obtained by considering this inequality in (6).If we choose δ = δ η (x), we can obtain the desired result.
Proof.For S η , we obtain is the space of the functions f , for which f , f ′ , and ) .Now, we define classical Peetre's-K functional as follows: where λ > 0.
. Then, we have for all η ∈ N, where Proof.For a given function g ∈ C 2 B [0, ∞), we have the following Taylor expansion Applying S η operator to Equation (7), we obtain Using the above inequality and Lemma 3, we obtain Thus, the proof is completed.
For f ∈ C B [0, ∞), the second modulus of continuity is explained by The relationship between Peetre's-K functional and the second modulus of continuity is given as follows: where the constant A is unaffected by the values of f and δ from [15].From ( 8) and ( 9),

Special Cases
In this section, we define Durrmeyer-Szász operators including confluent Bernoulli polynomials S B η and Durrmeyer-Szász operators including confluent Hermite polynomials S H η by selecting A(t) = t e t −1 and A(t) = e − t 2 2 in (2), respectively.
Theorem 5.For every x ∈ [0, ∞) and f ∈ C[0, ∞), Here, x + e 2 − 4e + 5 Proof.Using linearity of the operators S B η , we obtain By applying the Cauchy-Schwarz inequality to the last integral, we obtain Considering Cauchy-Schwarz inequality for summation and from Lemma 5, one can easily obtain where λ η (x) is given by (10).
can be obtained by considering this inequality in (11).If we choose δ = λ η (x), we achieve the desired result.

Approximation Properties for S H η
Choosing A(t) = e − t 2 2 in (2), then we obtain H (a,b) k . The confluent Hermite polynomials have as their generating function, where b / ∈ {. . . ,−1, 0}.The Szász-Durrmeyer operators including confluent Hermite polynomials are shown as Now, we give moments, central moments, and modulus of continuity for our operator including confluent Hermite polynomials.Lemma 6.For x ∈ [0, ∞), we obtain the moments for S H η as follows: Lemma 7.For every x ∈ [0, ∞) and by Lemma 6, the following identities verify where Proof.From the linearity of the operators S H η , we obtain For integration, we apply the Cauchy-Schwarz inequality and obtain Examining Cauchy-Schwarz disparity in summation and from Lemma 7, one can easily obtain where γ η (x) is given by (12).
can be obtained by considering this inequality in (13).If we choose δ = γ η (x), we obtain the desired result.

Graphical Analysis
In this section, we will examine the approximations of both Durrmeyer-type Szász operators and the newly defined confluent Szász-Durrmeyer operators to a function f .
Let the function f be Then, we plot the convergence of the newly constructed S η confluent Szász-Durrmeyer operators and Z η Durrmeyer-type Szász operators [4] to the function f in Figure 1 for A(t) = 1.In Figure 1 1000 , we show the error estimation of confluent Szász-Durrmeyer operators S η via the way of the modulus of continuity in Table 1.

Conclusions
In this study, Durrmeyer-type generalization of confluent Szász operators is constructed.The central moments of the newly constructed operators S η are obtained.Furthermore, the rate of convergence is investigated by using the modulus of continuity and Peetre's K-functional.The relationship between the newly constructed operators with B
k are given, respectively.Finally, the convergence of the confluent Szász-Durrmeyer operators S η and the classical Szász-Durrmeyer operators Z η to the selected functions are illustrated.The comparison of convergence is given by numerical examples.

Table 1 .
Error approximation for S η by using the modulus of continuity.