On the study of modified (p, q)- Bernstein polynomials and their applications

*Many researchers in different fields are keenly interested in the study of Bernstein polynomials from the day which it was invited to today. These studies involve both pure and applied branches as mathematics, statistics, numerical analysis, CAGD. The n th Bernstein polynomials, named after Bernstein (1912), are defined as Bk,n(x) = ( n k ) x(1 − x), where ( n k ) = n! k!(n−k)! for k ≤ n and x ∈ [0, 1]. After, about a century later, definition of the Bernstein polynomials, the generating function of these polynomials was obtained by Acikgoz and Araci (2010) as follows: ∑ Bk,n(x) tn n! = (xt)k k! e n=0 , t ∈ C. Identifying the generating functions is of major importance in mathematics and its fields such as number theory, combinatorics and so on. By using the generating function, the many Bernstein polynomials’ properties were obtained. Also, the generating function played important role between Bernstein polynomials and special numbers and polynomials and provided to arise new type definitions based on q-calculus (Acikgoz et al., 2010; 2012; Kim et al., 2010; Simsek, 2013; Simsek, 2017). * Corresponding Author. Email Address: agyuz@gantep.edu.tr (E. Agyuz) https://doi.org/10.21833/ijaas.2018.01.008 2313-626X/© 2017 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Let 0 < q < p ≤ 1. The (p, q)-analogue of n is defined by [n]p,q = pn−qn p−q . When p = 1 reduces to the qanalogue of n and also when q → p = 1 reduces to the ordinary integers (Chakrabarti and Jagannathan, 1991; Duran, 2016; Sadjang, 2013). (p, q)integers are generalization of qintegers such that we can write [n]p,q as below: [n]p,q = p [n]q p If we take p = 1, we obtain q-integers but the opposite is not true. So, we can not derive [n]p,q by the aid of qintegers (Gupta, 2016). Recently, the Bernstein polynomials have been moved into (p, q)calculus and are studied by many researchers. Mursaleen et al. (2015a) described (p, q) -Bernstein polynomials. The (p, q) -Bernstein polynomials are defined as below: Bk,n(x, p, q) = [ n k ] p,q p (k 2 )−(n2)xk(1 − x)p,q n−k where 0 < q < p ≤ 1 and (1 − x)p,q n−k = ∏ (p − n−k−1 j=0 xq). In addition, they gave Bernstein-Schurer operators, Bernstein-Stancu operators, Bernstein Kantorovich operators depend on (p, q)integers (Mursaleen et al., 2015a; 2015b; 2015c). Khan and Lobiyal (2015) obtained Lupaş Bernstein polynomials on (p, q)integers and gave their some properties with related to parametric curves. Acar et al. (2016) proposed Kantorovich modifications of (p, q)Bernstein operators and derived their rate of convergence, uniform convergence and approximations behaviors. Erkan Agyuz, Mehmet Acikgoz/ International Journal of Advanced and Applied Sciences, 5(1) 2018, Pages: 61-65 62 We will explore in the next chapter that Bk,n p,q (x) polynomials have some properties such as generating function, recurrence relations, derivative property and an identity with related to two important elements of special polynomials and numbers.


Introduction
*Many researchers in different fields are keenly interested in the study of Bernstein polynomials from the day which it was invited to today. These studies involve both pure and applied branches as mathematics, statistics, numerical analysis, CAGD.
The n th Bernstein polynomials, named after Bernstein (1912), are defined as After, about a century later, definition of the Bernstein polynomials, the generating function of these polynomials was obtained by  as follows: , t ∈ ℂ.
Identifying the generating functions is of major importance in mathematics and its fields such as number theory, combinatorics and so on. By using the generating function, the many Bernstein polynomials' properties were obtained. Also, the generating function played important role between Bernstein polynomials and special numbers and polynomials and provided to arise new type definitions based on q-calculus 2012;Kim et al., 2010;Simsek, 2013;Simsek, 2017).
When p = 1 reduces to the q-analogue of n and also when q → p = 1 reduces to the ordinary integers (Chakrabarti and Jagannathan, 1991;Duran, 2016;Sadjang, 2013). (p, q)-integers are generalization of q-integers such that we can write [n] p,q as below: [n] p,q = p n−1 [n]q p If we take p = 1, we obtain q-integers but the opposite is not true. So, we can not derive [n] p,q by the aid of q-integers (Gupta, 2016).
Recently, the Bernstein polynomials have been moved into (p, q)-calculus and are studied by many researchers. Mursaleen et al. (2015a) described (p, q) -Bernstein polynomials. The (p, q) -Bernstein polynomials are defined as below: xq j ). In addition, they gave Bernstein-Schurer operators, Bernstein-Stancu operators, Bernstein Kantorovich operators depend on (p, q)-integers (Mursaleen et al., 2015a;2015b;2015c). Khan and Lobiyal (2015) obtained Lupaş Bernstein polynomials on (p, q)-integers and gave their some properties with related to parametric curves. Acar et al. (2016) proposed Kantorovich modifications of (p, q)-Bernstein operators and derived their rate of convergence, uniform convergence and approximations behaviors.
We will explore in the next chapter that k,n p,q (x) polynomials have some properties such as generating function, recurrence relations, derivative property and an identity with related to two important elements of special polynomials and numbers.

Main results
In this part, we see that the generating function gives us useful properties about k,n p,q (x) polynomials.
Furthermore, we investigate results of these polynomials under the (p, q)-calculus.
Definition 2.1: For x ∈ [0,1], k ≤ n and 0 < q < p ≤ 1, If we take p = 1, the k,n p,q (x) reduces to Kim's modified q-Bernstein polynomials. Also, as q → p = 1, the k,n p,q (x) reduces to ordinary Bernstein polynomials. Some cases of the k,n p,q (x) polynomials are shown in Fig. 1. Now, we show that k,n p,q (x) is described by linear combination of two k,n−1 p,q (x) polynomials as below: Corollary 2.2: For x ∈ [0,1], k ≤ n and 0 < q < p ≤ 1, we have Proof: By using definition of k,n p,q (x) and property of Binomial coefficients, we obtain, We show that k,n p,q (x) polynomials have symmetric property as below: Corollary 2.3: For x ∈ [0,1], k ≤ n and 0 < q < p ≤ 1, we have n−k,n p,q (1 − x) = k,n p,q (x) Proof: By substituting x → 1 − x and k → n − k into above equation, we get We see that n − k th k,n−k p,q (x) polynomials are obtained by the aid of two n + 1 th k,n+1 p,q (x) polynomials by means of at the following corollary: Corollary 2.4: For x ∈ [0,1], k ≤ n and 0 < q < p ≤ 1, we have Proof: Proof of this corollary follows from definition of k,n p,q (x).
The expression of below corollary is to go from the k − 1 modified polynomials to k modified polynomials n th degree.

Corollary 2.5:
Proof: By applying the definition of The desired result is shown.
In the next corollary, we give the identity by the aid of summation of (p, q)-integers and binomial expansion, respectively. Corollary 2.6: For x ∈ [0,1], k ≤ n and 0 < q < p ≤ 1, we have Proof: By using of definition of k,n p,q (x) and binomial formula, we obtain Therefore, we arrive at the desired result. We give the reflection of k,n p,q ( ) polynomials under derivative operator at the following corollary: Corollary 2.7: For ∈ [0,1], ≤ and 0 < < ≤ 1, we have Proof: Using the definition of derivative, it is seen that and after making some algebraic operations, we obtain ].
Therefore, the proof of corollary is completed.

Bernstein polynomials
Now, we give the generating function of our modified polynomials at the following theorem: This ends the proof.
The generating function is shown for special values of , and in Fig. 2.  Therefore, we obtain the desired result.
On the other hand, the generating function can be rewritten in the following form: By aid of the definition of generating function, we have By choosing = at the above equation, we get Combining Eq. (1) and Eq.
(2), we arrive the desired result. After the Taylor series generated for − − , we get ).
By applying the Cauchy product both sides of above equality, we obtain By comparison of coefficients of ! , the proof is completed.

Relations between modify ( , )-Bernstein polynomials and special polynomials
In this section, we obtain some equalities interested in the modify polynomials the Euler polynomials, the Bernoulli polynomials and the Stirling numbers of the second kind. where the generating functions of the Genocchi polynomials and the Euler numbers are defined as follows: If we arrange the above equality, we get Thereby, by applying the Cauchy product for above equality and then comparison the coefficients of ! , we complete the proof. where the generating function of the Bernoulli polynomials and the second kind Stirling numbers are defined at the following equalities: .
By applying the Cauchy product at above equation as in the previous corollary, the desired result is obtained.

Conclusion
In this study, we extended the Bernstein polynomials to new type (p, q)-Bernstein polynomials. We also obtained some useful properties such as the generating function, symmetric function, recurrence relations and equalities under the derivative operators for these polynomials. We plotted the graphs of these polynomials and their generating function. Furthermore, some of our results are generalizations results in Kim et al. (2010) and Simsek (2017). Because of the Bernstein polynomials are very useful in many different fields such as statistics, engineering and CAGD, these new polynomials can be used in the mentioned areas.