The 
 q
 -Chlodowsky and 
 q
 -Szasz-Durrmeyer Hybrid Operators on Weighted Spaces

The main aim of this article is to introduce a new type of 
 
 q
 
 -Chlodowsky and 
 
 q
 
 -Szasz-Durrmeyer hybrid operators on weighted spaces. To this end, we give approximation properties of the modified new 
 
 q
 
 -Hybrid operators. Moreover, in the weighted spaces, we examine the rate of convergence of the modified new 
 
 q
 
 -Hybrid operators by means of moduli of continuity. In addition, we derive Voronovskaja’s type asymptotic formula for the related operators.


Introduction
Polynomial approach and the classical approximation theory constitute a basic research area in applied mathematics. e development of the approximation theory plays an important role in the numerical solution of partial differential equations, data processing sciences, and many other disciplines. For example, it is widely used in geometric modelling in the aerospace and automotive industries to calculate approximate values with basic functions. Works in this field go back to the 18th century and still continue as a powerful tool in scientific calculations. Furthermore, it is used in civil engineering projects to analyze the energy efficiency and earthquake resistance data of different types of buildings in thermography calculations and earthquake engineering.
In this study, the q− analog type operator is defined. e studies regarding the q− analog type operator are as follows.
First, the q− Bernstein polynomials were produced by Phillips [14]. When q � 1 is used, the results are the same as for classical operators. However, new operators with different properties are obtained for q ≠ 1. Gupta [15] introduced and analyzed the approach characteristics of q-Durrmeyer operators. Gupta and Heping [16] identified the q-Durrmeyer operators and estimated the rate of convergence for continuous functions with the help of the moduli of continuity. In [2,17], Mahmudov defined the King-type q-Szász operators. He obtained the rate of convergence on weighted spaces and a Voronovskaya-type theorem for these operators. Some other studies based on classical q− theory are [17][18][19][20][21][22][23][24].
In light of the above studies, the following result has been obtained. For a real-valued function, f(u) defined on the interval [0, ∞), the operators n k�0 P n,k,q (u) for the general operator kernels have been studied by many authors. On the contrary, the operator n k�0 P n,k,q (u) has not been studied yet. erefore, we introduced the following operator: which is q-Chlodowsky and q-Szasz-Durrmeyer hybrid operators on weighted spaces. Here, P n,k,q (x) and S n,k,q (y) are defined as in (2) and b n is an increasing and positive sequence with properties lim n⟶∞ b n � ∞ and lim n⟶∞ (b n /[n] q ) � 0. In this article, we intend to study the approximation properties of the operator H n,q (h; u). We produced our study by making use of [25].
e important terms of q-analysis which are used in this paper are given below, see [26,27], Given value of q > 0 and n ∈ N, we define the q-integer [n] q by for 0 ≤ k ≤ n; we define the q-binomial coefficients n k q by e q-binomial can be written in the following forms: Exponential function e z has two q-analogs, see [26,27]: where e definite q− Jackson integrals and the q− improper integrals of the function h are defined by e series on the right-hand side in (12) is guaranteed to be convergent if the function h has the property |h(u)| < Cu α in a right neighborhood of u � 0 for C > 0, α > − 1.
e q− Gamma functions are defined in two ways. ese are For every B and u > 0, one has Especially, for any positive integer m, see [28]. e present note deals with the study of the q-Chlodowsky and q-Szasz-Durrmeyer hybrid operators on weighted spaces. Firstly, we estimate the moments for the H n,q (h; u) operators.
We also study the rate of convergence for these operators H n,q (h; u). Furthermore, definitions and some properties for weighted spaces are given. We guess the order of approximation by weighted Voronovskaya-type theorem.

Estimation Moments
Here, we will prove H n,q (t m ; u) for m � 0, 1, 2. By the definition of the q− Gamma function c q in (15), we have where, for m � 0, for m � 1, and for m � 2, Proof. Using (19), we obtain

Journal of Mathematics 3
Using (20) Using (20) and which completes the proof.
Proof. By Lemma 1, we have which completes the proof.

□
By simple calculations, we obtain here; since n ⟶ ∞ will be q n ⟶ 1 and [n] q � n, the following equation is written: where H n,q ( (t − u) 2 ; u ) appears to be constrained in the above inequality, and this result is available in [25]. Likewise, A.İzgi obtained the following result in [25]:

Approximation of H n,q (h; u) in Weighted Spaces
In this section, we use Gadjiev's Korovkin-type theorems on the weighted spaces [4,29]. Let B p be the set of all functions h over the real line, where M h is a positive constant depending on the function h. Now, let It is clear that C k p (R) ⊂ C p (R) ⊂ B p (R), where B p (R) is the linear normed space with the norm (36)

Lemma 3. H n,q (h; u) defined in (5) is a sequence of positive and linear operators that move
Proof. Taking advantage of equations (22) and (24), we have under the condition in (36) that completes the proof. □

Journal of Mathematics
Proof. From Lemma 1, we have according to (36), which completes the proof.

Main Results
Here, we estimate the rate of approximation of the H n,q (h; u) hybrid operators. e following theorem gives the rate of approximation of the sequence of H n,q (h; u) operators in terms of moduli of continuity of a function h ∈ C k , the moduli of continuity is defined as follows: where δ ⟶ 0. e weighted moduli of continuity is defined as follows: It is seen that they provide the characteristics of the continuity module. In what follows, lim δ⟶0 Λ n (h, δ) � 0, for every h ∈ C k ρ [0, b n ]. A similar definition can be found in [5,25].
from (33), and we have which completes the proof.

Conclusion
In this paper, the approximation properties and rate of convergence of q-Chlodowsky and q-Szasz-Durrmeyer hybrid operators in weighted spaces are investigated. For further research in this topic, it would be interesting to study whether the quality of the approximation pythagorean fuzzy set operators, q-statistical convergence operators, and q-complex operators directly influence the quality of the approximation of the characteristics. Some studies can be used for future research, such as [30][31][32][33][34].

Data Availability
All data generated or analyzed during this study are included in this published article. ey are cited at relevant places within the text as references.