On shape parameter α based approximation properties and q -statistical convergence of Baskakov-Gamma operators

: We construct a novel family of summation-integral type hybrid operators in terms of shape parameter α ∈ [ 0, 1 ] in this paper. Basic estimates, rate of convergence, and order of approximation are also studied using the Korovkin theorem and the modulus of smoothness. We investigate the local approximation ﬁndings for these sequences of positive linear operators utilising Peetre’s K-functional, Lipschitz class, and second-order modulus of smoothness.


Introduction
The theory of linear positive operators deals with question that arise in the approximate representation of an arbitrary function by the simplest one. In the recent past the operator theory is a growing and fascinating field of research in approximation theory with the advent of computer. Several researchers constructed new sequences of linear positive operators to investigate the rapidity of convergence and order of approximation in different function spaces in terms of several generating functions. Recently, Chen et al. [11] constructed a sequence of new linear positive operators known as the α-Berstein operators of order m: for g ∈ C[0, 1], m ∈ N and α ∈ [−1, 1], where p (α) 1,1 = y and p α m,i (y) These operators are restricted for the space of continuous functions only. To approximate the wider class than the class of continuous function, i.e., space of Lebesgue integrable functions, Mohiuddine et al. constructed Kantorovich-type of α-Bernstein operators [19] and Stancu-type α-Bernstein-Kantorovich operators [18]. Cai et al. [10] introduced a generalization of classical Bernstein operators based on shape parameter α ∈ [0, 1]. These operators are termed as α−Bernstein operators of degree m and defined as: where p (α) m,i (y) is defined by (2).
Remark 2. Note that, p α m,i in the relation (3) is called α-Berstein polynomials of order m and the binomial coefficients Later on, Aral and Erbay [6] introduced the parametric form of Baskakov-Durrmeyer operators as: The sequences (4) are restricted for the space of continuous functions only. Motivated by the above development, we construct a sequence of hybrid operators to approximate in a wider class, i.e., the space of Lebesgue integrable functions, as follows: where P (α) m,s (w) is given by (4) and the gamma function as: In the subsequent sections, we establish basic lemmas, rate of convergence, order of approximation locally and globally in terms of modulus of smoothness, Peetre's K-functional, second modulus of smoothness, Lipschitz space maximal function and weighted modulus of smoothness. Lastly, we study the q-statistical convergence. For more basic concepts and related articles we refer to see the published article [3,5,9,17,[21][22][23][24]26] 2. Basic Estimates and approximation Lemma 1. [6] For m ∈ N, the α-Baskakov operator has the following identities:
Proof. In view of Lemma 2, for e 0 = 1, we have For e 1 = t, we have Proof. In the light of linearity property of A * m,α (.; .) and Lemma (2), we get the desired Lemma (3).

Definition 1. The modulus of continuity for a uniformly continuous function f on
Also, we get . Proof. Taking into account the property (vi) of Theorem 4.1.4 [4], it is enough to show that From Lemma 2, we get A * m,α (e j ; w) → e j (w) for j = 0, 1, 2 when m → ∞. Which gives the prove of Theorem 1.  Proof. In the light of Lemma 2, Lemma 3 and Theorem 2, it is easy to obtain .
From Taylor series expansion, for any g ∈ C 2 B [0, ∞), we have On operating A m,α ( f ; w) in (8), we obtain Therefore in the view of (7) we get, Then Combining the equalities (9), (10) and (11), we see that which completes the proof.
Theorem 5. For any g ∈ C 2 B [0, ∞), there exists a positive number C satisfying the inequality where ξ z m is given by Theorem 4.
This gives the proof of Theorem 5.
For any fixed two real positive numbers s 1 and s 2 , the Lipschitz-class of functions [25] is defined by: with the positive constant C and 0 < β ≤ 1.
Using theḦolder's inequality with p 1 = 2 r and p 2 = 2 2−r , we have which gives the desired result.
Theorem 9. Suppose the operators A * m,α (.; .) acting from C k Proof. For the results of Theorem 9, we have to show that Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 25 January 2022 doi:10.20944/preprints202201.0383.v1 By using the Lemma 2, it is enough to show and we get A * m,α (e 2 ) − w 2 1+w 2 → 0 as m → ∞. Hence we get the result.
Here, we study the approximation of locally integrable functions belongs to C k 1+w 2 [0, ∞). Such type of result is investigated by Gadjiev [15].
In the light of Theorem 9 and for any m 2 > m, one has Take m 3 = max(m 1 , m 2 ) and taking (14) by values of I 1 , I 2 , I 3 we easily get Thus, the proof of Theorem 10 is completed.

q-Density and q-statistical convergence
Recently, the q-analog of density and statistical convergence are studied in [2]. Let E ⊆ N (the set of natural numbers). Then the q-density is defined by , q ≥ 1, where C 1 (q) = (c 1 nk (q k )) ∞ n,k=0 is the q-Cesàro matrix (see [1], [2]) defined by otherwise.
which is regular for q ≥ 1, where the q-integer (q > 0) of any positive integer n is defined by #{k ≤ n : q k−1 |η k − l| ≥ ε} = 0 and we write St q − lim η k = l.
Letting m → ∞, we have This gives the desired proof of Theorem 11.

Conflicts of Interest
The authors declare that they have no conflicts of interest.