#### Abstract

This paper deals with approximating properties of the newly defined -generalization of the genuine Bernstein-Durrmeyer polynomials in the case , which are no longer positive linear operators on . Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine -Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in , , the rate of approximation by the genuine -Bernstein-Durrmeyer polynomials is of order versus for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine -Bernstein-Durrmeyer for . This paper represents an answer to the open problem initiated by Gal in (2013, page 115).

#### 1. Introduction

In several recent papers, convergence properties of complex -Bernstein polynomials, proposed by Phillips [1], attached to an analytic function in closed disks, were intensively studied. Ostrovska [2, 3] and Wang and Wu [4, 5] have investigated convergence properies of in the case . In the case , the -Bernstein polynomials are no longer positive operators; however, for a function analytic in a disc , it was proved in [2] that the rate of convergence of to has the order (versus for the classical Bernstein polynomials). Moreover, Ostrovska [3] obtained Voronovskaya-type theorem for monomials. If , then qualitative Voronovskaja-type theorem and saturation results for complex -Bernstein polynomials were obtained by Wang and Wu [4]. Wu [5] studied saturation of convergence on the interval for the -Bernstein polynomials of a continuous function for arbitrary fixed .

Genuine Bernstein-Durrmeyer operators were first considered by Chen [6] and Goodman and Sharma [7] around 1987. In recent years, the genuine Bernstein-Durrmeyer operators have been investigated intensively by a number of authors. Among the many papers written on the genuine Bernstein-Durrmeyer operators, we mention here only the ones by Gonska et al. [8], Parvanov and Popov [9], Sauer [10], Waldron [11], and the book of PÄƒltÄƒnea [12].

On the other hand, Gal [13] obtained quantitative estimates of the convergence and of the Voronovskaja-type theorem in compact disks, for the complex genuine Bernstein-Durrmeyer polynomials attached to analytic functions. Besides, in other very recent papers, similar studies were done for complex Bernstein-Durrmeyer operators in Anastassiou and Gal [14], for complex Bernstein-Durrmeyer operators based on Jacobi weights in Gal [15], for complex genuine -Bernstein-Durrmeyer operators () by Mahmudov [16], and for other kinds of complex Durrmeyer operators in Mahmudov [17] and Gal et al. [18]. It should be stressed out that study of -Durrmeyer-type operators () in the real case was first initiated by Derriennic [19].

Also, for the case , exact quantitative estimates and quantitative Voronovskaja-type results for complex -Lorentz polynomials, -Stancu polynomials [20], -Stancu-Faber polynomials, -Bernstein-Faber polynomials, -Kantorovich polynomials [21], -SzÃ¡sz-Mirakjan operators [22] obtained by different researchers are collected in the recent book of Gal [23]. In this book the definition and study of complex -Durrmeyer-kind operators for presented an open problem. This paper presents a positive solution to this problem.

In this paper we define the genuine -Bernstein-Durrmeyer polynomials for . Note that similar to the -Bernstein operators the genuine -Bernstein-Durrmeyer operators in the case are not positive operators on . The lack of positivity makes the investigation of convergence in the case essentially more difficult than that for . We present upper estimates in approximation and we prove the Voronovskaja-type convergence theorem in compact disks in , centered at origin, with quantitative estimate of this convergence. These results allow us to obtain the exact degrees of approximation by complex genuine -Bernstein-Durrmeyer polynomials. Our results show that approximation properties of the complex genuine -Bernstein-Durrmeyer polynomials are better than approximation properties of the complex Bernstein-Durrmeyer polynomials considered in [13].

#### 2. Main Results

We begin with some notations and definitions of -calculus; see, for example, [24, 25]. Let . For any , the -integer is defined by and the -factorial is defined by For integers , the -binomial is defined by For we obviously get , , and . Moreover

For fixed , , we denote the -derivative of by

The -analogue of integration in the interval (see [24]) is defined by Let be a disc in the complex plane . Denote by the space of all analytic functions on . For we assume that for all . The norm . We denote for all .

*Definition 1. *For , the genuine -Bernstein-Durrmeyer operator is defined as follows:
where for the sum is empty; that is, it is equal to .

are linear operators reproducing linear functions and interpolating every function at and . The genuine -Bernstein-Durrmeyer operators are positive operators on for , and they are not positive for . As a consequence, the cases and are not similar to each other regarding the convergence. For and we recapture the classical () genuine Bernstein-Durrmeyer polynomials.

We start with the following quantitative estimates of the convergence for complex -Bernstein-Durrmeyer polynomials attached to an analytic function in a disk of radius and center .

Theorem 2. *Let , , and . Then for all one has
*

Theorem 2 says that, for functions analytic in , , the rate of approximation by the genuine -Bernstein-Durrmeyer polynomials () is of order versus for the classical genuine Bernstein-Durrmeyer polynomials; see [13].

The Voronovskaja theorem for the real case with a quantitative estimate is obtained by Gonska et al. [26] in the following form: and, for all , . For the complex genuine -Bernstein-Durrmeyer () a quantitative estimate is obtained by Gal [13] () and Mahmudov [16] () in the following form: and, for all , .

To formulate and prove the Voronovskaja-type theorem with a quantitative estimate in the case we introduce a function .

Let and let . For , we define And, for ,

The next theorem gives Voronovskaja-type result in compact disks; for complex -Bernstein-Durrmeyer polynomials attached to an analytic function in , and center in terms of the function .

Theorem 3. *Let , , and . The following Voronovskaja-type result holds:
**
For all , .*

Now we are in position to prove that the order of approximation in Theorem 2 is exactly versus for the classical genuine Bernstein-Durrmeyer polynomials; see [13].

Theorem 4. *Let , , and . If is not a polynomial of degree â‰¤1, the estimate,
**
holds, where the constant depends on , , and but is independent of .*

From Theorem 3 we conclude that, for , in and therefore . Furthermore, we have the following saturation of convergence for the genuine -Bernstein-Durrmeyer polynomials for fixed .

Theorem 5. *Let , . If a function is analytic in the disc , then for infinite number of points having an accumulation point on if and only if is linear.*

The next theorem shows that , is continuous in the parameter for , .

Theorem 6. *Let and . Then, for any ,
**
uniformly on .*

#### 3. Auxiliary Results

The -analogue of beta function for (see [24]) is defined as Since we consider the case , we need to use as follows: Also, it is known that For , we have Thus, we get the following formula for : Note that, for , we have

Lemma 7. * is a polynomial of degree less than or equal to and
*

*Proof. *From (20) it follows that
Now using
where , , are the constants independent of , we get
Since is a polynomial of degree less than or equal to and , , it follows that is a polynomial of degree less than or equal to .

Lemma 8. *The numbers , given by (24), enjoy the following properties:
*

Also, the following lemma holds.

Lemma 9. *For all the identity,
**
holds.*

*Proof. *It follows from end points interpolation property of and . Indeed

Lemma 9 implies that for all and we have

For our purpose first we need a recurrence formula for .

Lemma 10. *For all and one has
*

*Proof. *By simple calculation we obtain (see [27])
It follows that
which implies the recurrence in the statement.

Let Using the recurrence formula (30) we prove two more recurrence formulas.

Lemma 11. *For all and one has
**
where
*

*Proof. *From the recurrence formula in Lemma 10, for all , we get
where
Again by simple calculation we obtain
where and can be simplified as follows:

Lemma 12. *Let and . The function has the following representation:
*

*Proof. *Using the following identity:
we get
where .

#### 4. Proofs of the Main Results

Firstly we prove that . Indeed denoting with , by the linearity of , we have and it is sufficient to show that, for any fixed and with , we have . But this is immediate from , the norm being defined as , and from the inequality valid for all , where Therefore we get as and .

*Proof of Theorem 2. *From the recurrence formula (34) and the inequality (29) for we get
It is known that, by a linear transformation, the Bernstein inequality in the closed unit disk becomes
which, combined with the mean value theorem in complex analysis, implies
for all , where is a complex polynomial of degree â‰¤. It follows that
By writing the last inequality for , we easily obtain, step by step, the following:
It follows that

The second main result of the paper is the Voronovskaja-type theorem with a quantitative estimate for the complex version of genuine -Bernstein-Durrmeyer polynomials.

*Proof of Theorem 3. *By Lemma 11 we have
where
It follows that
for all , , and . Equation (54) implies that for
By writing the last inequality for , we easily obtain, step by step, the following:

*Proof of Theorem 4. *For all and we get
It follows that
Because by hypothesis is not a polynomial of degree in , it follows . Indeed, assuming the contrary it follows that for all ; that is, for all . Thus and is linear, which is a contradiction with the hypothesis.

Now, by Theorem 3, we have
Consequently, there exists (depending only on and ) such that for all we have