摘要：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 Z p $\mathbb{Z} _{p}$ , 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 Z p $\mathbb{Z} _{p}$ . Also we deduce a fermionic p-adic q-integral representation of products of weighted q-Bernstein polynomials of different degrees n 1 , n 2 , … $n_),n_,,\ldots $ on Z p $\mathbb{Z} _{p}$ 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.
关键词：Genocchi numbers and polynomials ; q -Genocchi numbers and polynomials ; weighted q -Genocchi numbers and polynomials ; Bernstein polynomials ; q -Bernstein polynomials ; weighted q -Bernstein polynomials
