On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials

and Applied Analysis 3 Recently, Varma et al. 5 constructed linear positive operators including Brenke-type polynomials. Brenke-type polynomials 6 have generating functions of the form


Introduction
The approximation theory, which is concerned with the approximation of functions by simpler calculated functions, is a branch of mathematical analysis. In 1885, Weierstrass identified the set of continuous functions on a closed and bounded interval through uniform approximation by polynomials. Later, Bernstein gave the first impressive example for these polynomials.
In 1953, Korovkin  where n ∈ N, x ≥ 0, and f ∈ C 0, ∞ whenever the above sum converges. Many researchers have dealt with the generalization of Szasz operators in a natural way. Later, Jakimovski and Leviatan 3 presented a generalization of Szasz operators with Appell polynomials. Let g z ∞ k 0 a k z k a 0 / 0 be an analytic function in the disc|z| < R R > 1 and assume that g 1 / 0. The Appell polynomials p k x have generating functions of the form Under the assumption p k x ≥ 0 for x ∈ 0, ∞ , Jakimovski and Leviatan introduced the linear positive operators P n f; x via and gave the approximation properties of these operators. p k x t k , |t| < R. 1.4 Using the following assumptions:

1.5
Ismail investigated the approximation properties of linear positive operators given by and have the following explicit expression: Using the following assumptions ii B : 0, ∞ −→ 0, ∞ , iii 1.7 and the power series 1.8 converge for |t| < R R > 1 ,

1.10
Varma et al. introduced the following linear positive operators involving the Brenke-type polynomials where x ≥ 0 and n ∈ N.
In this case, the operators 1.11 resp., 1.7 reduce to the operators given by 1.3 resp., 1.2 .
Remark 1.5. Let B t e t and A t 1. We meet again the Szasz operators 1.1 .
In this paper, our aim is to construct linear positive operators by using Boas-Bucktype polynomials including the Brenke-type polynomials, Sheffer polynomials, and Appell polynomials with special cases. Boas-Buck-type polynomials 7 have generating functions of the type where A, B, and H are analytic functions

1.13
We will restrict ourselves to the Boas-Buck-type polynomials satisfying iii 1.12 and the power series 1.13 converge for |t| < R R > 1 .

1.14
Now, given the above restrictions, we present a new form of linear positive operators with Boas-Buck-type polynomials as follows: where x ≥ 0 and n ∈ N.
It is obvious that one can get the operators 1.3 from the operators 1.15 . In addition, if we choose A t 1, we meet again the well-known Szasz operators 1.1 .
The paper is divided into three sections. Following the introduction, Section 2 is devoted to obtain qualitative and quantitative results for the operators 1.15 . In the last section, we give some significant illustrations with the help of Laguerre, Charlier, and Gould-Hopper polynomials for the operators 1.15 . Moreover, we give a Voronovskaya-type theorem for the operators including Gould-Hopper polynomials.
Abstract and Applied Analysis 5

Approximation Properties of B n Operators
In this section, with the help of well-known Korovkin's theorem, we get approximation results by means of B n linear positive operators. Next, we present quantitative results for estimating the error of approximation using the classical approach and the second modulus of continuity.
Proof. From the generating functions of the Boas-Buck-type polynomials given by 1.12 , we obtain

2.2
With regard to these equalities, we get the assertions of the lemma.
Let us define the class of E as follows: In order to estimate the rate of convergence, we will give some definitions and lemmas.

Lemma 2.5 Gavrea and Raşa 9 .
Let g ∈ C 2 0, a and K n n≥0 be a sequence of linear positive operators with the property K n 1; x 1. Then, Lemma 2. 6 Zhuk 10 . Let f ∈ C a, b and h ∈ 0, b − a /2 . Let f h be the second-order Steklov function attached to the function f. Then, the following inequalities are satisfied:

2.11
Proof. Using the linearity property of B n operators, one can write Applying Lemma 2.1, we obtain the equality stated in the lemma.
Generally, we use the modulus of continuity and second modulus of continuity to obtain quantitative error estimation for convergence by linear positive operators. Now, we will calculate the rate of convergence in the following two theorems. Theorem 2.8. Let f ∈ C 0, ∞ ∩ E. B n operators verify the following inequality:

2.14
Proof. Making use of Lemma 2.1 and the property of modulus of continuity, we deduce

Abstract and Applied Analysis
Taking into account the Cauchy-Schwarz inequality and then by using Lemma 2.7, we get

2.16
Considering the last inequality in 2.15 , we obtain where ϑ n x is given by 2.14 . In inequality 2.17 , by choosing δ ϑ n x , we get the desired result. Theorem 2.9. For f ∈ C 0, a , the following estimate Proof. Let f h be the second-order Steklov function attached to the function f. With regard to the identity B n 1; x 1, we have Taking account of the fact that f h ∈ C 2 0, a , it follows from Lemma 2.5 If one combines Landau inequality with Lemma 2.6, we can write

Examples
It is clear that Laguerre polynomials are Boas-Buck-type polynomials. Note that when x ∈ −∞, 0 , L α k x are positive. For ensuring the restrictions 1.14 and the assumptions 2.4 , we have to modify the generating functions 3.1 as follows: With the help of generating functions 3.3 , we find the following linear positive operators including Laguerre polynomials from the operators 1.15 where α > −1 and x ∈ 0, ∞ .

Remark 3.2.
It is worthy to note that we obtain new linear positive operators different from the one given in 4 .  x, h t k k! 3.9