A note on the values of weighted q-Bernstein polynomials and weighted q-Genocchi numbers

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 (or called q-Bernstein polynomials with weight α) and weighted q-Genocchi numbers (or called q-Genocchi numbers with weight α and β). We apply the method of generating function and p-adic q-integral representation on Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{Z} _{p}$\end{document}, 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 α and β. Furthermore, we derive an integral representation of weighted q-Bernstein polynomials of degree n based on Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{Z} _{p}$\end{document}. Also we deduce a fermionic p-adic q-integral representation of products of weighted q-Bernstein polynomials of different degrees n1,n2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n_{1},n_{2},\ldots $\end{document} on Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{Z} _{p}$\end{document} and show that it can be in terms of q-Genocchi numbers with weight α and β, which yields a deeper insight into the effectiveness of this type of generalizations. We derive a new generating function which possesses a number of interesting properties which we state in this paper.


Introduction
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 q-exponential functions, and Gauss, who discovered q-binomial formula. The systematic development of q-calculus begins from FH Jackson who  reintroduced the Euler-Jackson q-difference operator (Jackson, ). One of the important branches of q-calculus is q-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 joined in the link of p-adic integral and developed. In spite of them being already one hundred years old, these special numbers and polynomials, for instance, q-Bernstein polynomials, q-Genocchi numbers and polynomials, 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 L functions. 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 points of view along with some generalizations and modifications by a number of researchers. By using the p-adic invariant q-integral on Z p , Kim [] constructed p-adic Bernoulli numbers and polynomials with weight α. He also gave the identities on the q-integral representation of the product of several q-Bernstein polynomials and constructed a link between q-Bernoulli polynomials and q-umbral calculus (cf. [, ]). Our aim of this paper is also to show that a fermionic p-adic q-integral representation of products of weighted q-Bernstein polynomials of different degrees n  , n  , . . . on Z p can be written in terms of q-Genocchi numbers with weight α and β.
Suppose that p is chosen as an odd prime number. Throughout this paper, we make use of the following notations: Z p denotes 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 ∪ {}. The normalized p-adic absolute value is defined by |p| p =  p . When one mentions q-extension, q can be variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ C p . If q ∈ C, we assume |q| < . If q ∈ C p , we assume |q -| p < p For α, k, n ∈ N * and x ∈ [, ], Kim et al. defined weighted q-Bernstein polynomials as follows:

it turns out to be the classical Bernstein polynomials (see [] and []).
The In [], for n ∈ N * , modified q-Genocchi numbers with weight α and β are defined by Araci et al. as follows: In the case, for x = , we have g n,q that are called q-Genocchi numbers with weight α and β.
In [], 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 obtain 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.

On the weighted q-Genocchi numbers and polynomials
In this part, we will give some arithmetical properties of weighted q-Genocchi polynomials by using the techniques of p-adic integral and the method of generating functions. Thus, by utilizing the definition of weighted q-Genocchi polynomials, we have Thus we state the following theorem.
Theorem  Suppose n, α, β ∈ N * . Then we have by using the umbral (symbolic) convention (g (α,β) q By expression of (.), we get Consequently, we obtain the following theorem.
Theorem  The following is true.
From expression of (.) and Theorem , we get the following theorem.
Theorem  The following identity holds: with the usual convention about replacing (g For n, α ∈ N, by Theorem , we note that Consequently, we state the following theorem.
Theorem  Suppose n ∈ N. Then we have From expression of Theorem  and (.), we easily see that Thus, we obtain the following theorem.

Theorem  The following identity
is true.
Suppose n, α ∈ N. By expression of Theorem  and Theorem , we get For (.), we obtain the corollary as follows.
Corollary  Suppose n, α ∈ N * . Then we have

Novel identities on the weighted q-Genocchi numbers
In this section, we develop modified 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 (.), Kim et al. get the symmetry of q-Bernstein polynomials weight α as follows: Thus, from Corollary , (.) and (.), 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 (.) and (.), we state the following theorem. Suppose n  , n  , k ∈ N * and α ∈ N with n  + n  > k. It yields Therefore, we obtain the following theorem.