A note on the values of the weighted q-Bernstein polynomials and modified q-Genocchi numbers with weight alpha and beta via the p-adic q-integral on Zp

The rapid development of q-calculus has led to the discovery of new generalizations of Bernstein polynomials and Genocchi polynomials involving q-integers. The present paper deals with weighted q-Bernstein polynomials and q-Genocchi numbers with weight alpha and beta. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of Bernstein polynomials and q-Genocchi numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between q-Genocchi numbers and polynomials with weight alpha and beta. Furthermore, we derive an integral representation of weighted q-Bernstein polynomials of degree n on Zp. Also we deduce a fermionic p-adic q-integral representation of product weighted q-Bernstein polynomials of different degrees n1,n2,...on Zp and show that it can be written with q-Genocchi numbers with weight alpha and beta which yields a deeper insight into the effectiveness of this type of generalizations. Our new generating function possess a number of interesting properties which we state in this paper.


Introduction, Definitions and Notations
The q-calculus theory is a novel theory that is based on finite difference re-scaling. First results in q-calculus belong to Euler, who discovered Euler's Identities for qexponential functions and Gauss, who discovered q-binomial formula. The systematic development of q-calculus begins from F. H. Jackson who 1908 reintroduced the Euler Jackson q-difference operator (Jackson, 1908). One of important branches of q-calculus is q-type special orthogonal polynomials. Also p-adic numbers were invented by Kurt Hensel around the end of the nineteenth century and these two branches of number theory jointed with the link of p-adic q-integral and developed. In spite of their being already one hundred years old, these special numbers and polynomials, for instance q-Bernstein numbers and polynomials, q-Genocchi numbers and polynomials and etc. are still today enveloped in an aura of mystery within the scientific community. The p-adic integral was used in mathematical physics, for instance, the functional equation of the q-zeta function, q-stirling numbers and q-Mahler theory of integration with respect to the ring Z p together with Iwasawa's p-adic q-L functions. Professor T. Kim [29], also studied on p-adic interpolation functions of special orthogonal polynomials. In during the last ten years, the q-Bernstein polynomials and q-Genocchi polynomials have attracted a lot of interest and have been studied from different angles along with some generalizations and modifications by a number of researchers. By using the p-adic invariant q-integral on Z p , Professor T. Kim in [26], constructed p-adic Bernoulli numbers and polynomials with weight α. After Seo and first author in [9], extended Kim's method for q-Genocchi numbers and polynomials and also they defined q-Genocchi numbers and polynomials with weight α and β. Our aim of this paper is to show that a fermionic p-adic q-integral representation of product weighted q-Bernstein polynomials of different degrees n 1 , n 2 , · · · on Z p can be written with q-Genocchi numbers with weight α and β.
Let p be a fixed odd prime number. Throughout this paper we use the following notations. By Z p we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Q p denotes the field of p-adic rational numbers, and C p denotes the completion of algebraic closure of Q p . Let N be the set of natural numbers and N * = N ∪ {0}. The p-adic absolute value is defined by |p| p = 1 p . In this paper we assume |q − 1| p < 1 as an indeterminate. In [23][24][25], let U D (Z p ) be the space of uniformly differentiable functions on Z p . For f ∈ U D (Z p ), the fermionic p-adic q-integral on Z p is defined by T. Kim: For α, k, n ∈ N * and x ∈ [0, 1], T. Kim et al. defined weighted q-Bernstein polynomials as follows: n−k q −α , (for detail, see [3,27,33,34]).
In (1.2), we put q → 1 and α = 1, n−k and we obtain the classical Bernstein polynomials (see [1], [2]), where, [x] q is a q-extension of x which is defined by In previous paper [8], for n ∈ N * , modified q-Genocchi numbers with weight α and β are defined by Araci et al. as follows: In the special case, are called the q-Genocchi numbers with weight α and β.
In [8], for α ∈ N * and n ∈ N, q-Genocchi numbers with weight α and β are defined by Araci et al. as follows: In this paper, we obtained some relations between the weighted q-Bernstein polynomials and the modified q-Genocchi numbers with weight α and β. From these relations, we derive some interesting identities on the q-Genocchi numbers with weight α and β.

2.
On the q-Genocchi numbers and polynomials with weight α and β By the definition of q-Genocchi polynomials with weight α and β, we easily get Therefore, we obtain the following Theorem: Moreover, By expression of (1.3), we get Consequently, we obtain the following Theorem: Theorem 2. The following From expression of (2.2) and Theorem 1, we get the following Theorem: Theorem 3. The following identity holds g (α,β) 0,q = 0, and q −α q α g (α,β) with the usual convention about replacing g (α,β) q n by g (α,β) n,q .
For n, α ∈ N, by Theorem 3, we note that n,q , if n > 1. Consequently, we state the following Theorem: From expression of Theorem 2 and (2.3), we easily see that n+1,q −1 (2) . Thus, we obtain the following Theorem.
Theorem 5. The following identity is true.
Let n, α ∈ N. By expression of Theorem 4 and Theorem 5, we get For (2.5), we obtain corollary as follows:

Novel identities on the weighted q-Genocchi numbers
In this section, we develop modifed q-Genocchi numbers with weight α and β, namely, we derive interesting and worthwhile relations in Analytic Number Theory.
For x ∈ Z p , the p-adic analogues of weighted q-Bernstein polynomials are given by By expression of (3.1), Kim et. al. get the symmetry of q-Bernstein polynomials weight α as follows: n−k,n 1 − x, q −1 , (for detail, see [27]). Thus, from Corollary 1, (3.1) and (3.2), we see that For n, k ∈ N * and α ∈ N with n > k, we obtain Let us take the fermionic p-adic q-integral on Z p on the weighted q-Bernstein polynomials of degree n as follows: Consequently, by expression of (3.3) and (3.4), we state the following Theorem: Theorem 6. The following identity holds Let n 1 , n 2 , k ∈ N * and α ∈ N with n 1 + n 2 > 2k. Then, we get Therefore, we obtain the following Theorem: Theorem 7. For n 1 , n 2 , k ∈ N * and α, β ∈ N with n 1 + n 2 > 2k, we have By using the binomial theorem, we can derive the following equation.
Thus, we can obtain the following Corollary: Corollary 2. For n 1 , n 2 , k ∈ N * and α ∈ N with n 1 + n 2 > 2k, we have For ξ ∈ Z p and s ∈ N with s ≥ 2, let n 1 , n 2 , ..., n s , k ∈ N * and α ∈ N with s l=1 n l > sk. Then we take the fermionic p-adic q-integral on Z p for the weighted q-Bernstein polynomials of degree n as follows: , if k = 0.
So from above, we have the following Theorem: Theorem 8. For s ∈ N with s ≥ 2, let n 1 , n 2 , ..., n s , k ∈ N * and α ∈ N with s l=1 n l > sk. Then we have Therefore, from (3.6) and Theorem 8, we get interesting Corollary as follows: Corollary 3. For s ∈ N with s ≥ 2, let n 1 , n 2 , ..., n s , k ∈ N * and α ∈ N with s l=1 n l > sk. We have , if k = 0.