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

Abstract

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 alpha) and weighted q-Genocchi numbers (or called q-Genocchi numbers with weight alpha and beta). We apply the method of generating function and p-adic q-integral representation on 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 alpha and beta. Furthermore, we derive an integral representation of weighted q-Bernstein polynomials of degree n based on 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), ... on Z(p) and show that it can be in terms of q-Genocchi numbers with weight alpha and beta, 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.