A note on the convergence of Phillips operators by the sequence of functions via q-calculus

1 Introduction and auxiliary results

The Szász positive linear operators are discussed in this article, and approximation properties are given by incorporating the parametric nonnegative variations. We refer to C [ 0 , ∞ ) as the class of all continuous functions defined on the interval [ 0 , ∞ ) . Szász [1] introduced the following positive linear operators in 1950 for functions f ∈ C [ 0 , ∞ ) :

The operators in (1.1) are more powerful and deal with the generalized approximation properties rather than the earlier Bernstein polynomials obtained by Bernstein in 1912 [2]. Sucu [3] recently introduced the Szász operators (1.1) by applying the exponential generating functions in Dunkl form with a new real parameter ϱ on [ 0 , ∞ ) . The following new variant of Szász-Dunkl operators given by Içöz and Çekim [4] for ϱ > 1 2 , namely the q -Szász-Dunkl form, is used to discuss the approximation of Szász operators (1.1):

for ϱ > 1 2 , x ≥ 0 , 0 < q < 1 , and f ∈ C [ 0 , ∞ ) . While for ϱ > − 1 2 , Cheikh et al. [5] utilized the q -Hermite polynomials and introduced the definitions of q -Dunkl analogue for the exponential generating function for x ∈ [ 0 , ∞ ) , and they also investigated the recursion formulae as follows:

where N = { 1 , 2 , 3 , 4 , … } and J = N ∪ { 0 } . Moreover, a special formula of γ ϱ , q ( τ ) is mentioned as follows:

and some basic calculations for τ ∈ N ∪ { 0 } in the Dunkl q -integer form are

The q -integer [ τ ] q and q -factorial [ τ ] q ! , respectively, are defined as follows:

In the field of approximation theory, the q -calculus has gained a lot of traction and has a lot of applications. It is capable of dealing with all aspects of science, including mathematical science, physical science, computer science, and other disciplines. Basic hypergeometric functions, number theory, harmonic analysis, orthogonal polynomials, combinatorics, quantum theory, mechanics, theory of relativity, and other mathematical topics are among the many applications and individuals linked with the q -calculus (see [6, 7]). Most recently, the notion of q -calculus has been used in summability [8] and approximation theory [9]. By introducing exponential generating functions in multiple extended formulations and having additional applications, the Szász operators have recently been associated with more improvement than the classical operators, for which we refer the reader to see [10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24] and for some related concepts, see [25,26,27].

In this article, we apply the Dunkl method of generating exponential functions in q -calculus to investigate the approximation capabilities of Phillips operators. The Korovkin-type theorem in weighted spaces is obtained by focusing on functions with characteristics of modulus of continuity. We successfully calculate the order of approximation and investigate some direct theorems by applying the Lipschitz class and Peetre’s K -functional. Next, we also discuss here the Voronovskaja-type approximation theorem.

Let { ξ τ , q ( x ) } τ ≥ 1 be a sequence of continuous functions defined on semi-axis R + = [ 0 , ∞ ) such that

with the notations

We suppose C ζ [ 0 , ∞ ) = { f such that f ∈ C [ 0 , ∞ ) and f ( t ) = O ( t ζ ) , t → ∞ } , x ∈ [ 0 , ∞ ) , ζ > τ , τ ∈ N ∪ { 0 } , and ϱ ≥ − 1 2 , we define, for all f ∈ C ζ [ 0 , ∞ ) ,

where 0 ≤ γ ≤ λ , T = [ τ ] q t + γ [ τ ] q + λ , and

In the case of γ = λ = 0 , we simply obtain that our operators (1.10) are reduced to the operators studied in [28]. In our modifications of generalized operators, we approach the better observations and more appropriate approximation results rather than [28]. In addition, we can say that these types of approximation results are the extended form of the most recent published article by [7,28,29]. If q = 1 , then [29] is reduced to [28], and if γ = λ = 0 , then our operators are reduced to [28], while the article [29] is the generalized formulation of [7].

To estimate the moments, we have successfully used some basic definitions of q -gamma functions.

May [7] discovered an inversion formula for the semigroups of positive linear operators in 1954, and the approximation features of Phillips operators were recently examined using the Dunkl generalization of an exponential form, which was executed in [29]. Introducing the Dunkl generalization of q -exponential form [28], which incorporates the generalized approximation properties of Phillips operators for functions f ∈ C [ 0 , ∞ ) rather than those given in [7,29], has recently improved the approximation results by Phillips operators. Rather than the published articles [7,28,29,31,32], our generalization of Phillips operators provides more relevant and modified convergence features in quantum calculus. Furthermore, in the case of γ = λ = 0 , we find that our new operators simply reduce to the most recently studied operators in [28].

2 Estimation of the associated moments of operators P τ , q ∗