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The rapid development of q-calculus has led to the discovery of newgeneralizations of Bernstein polynomials and Genocchi polynomials involvingq-integers. The present paper deals with weighted q-Bernstein polynomials andq-Genocchi numbers with weight alpha and beta. We apply the method ofgenerating function and p-adic q-integral representation on Zp, which areexploited to derive further classes of Bernstein polynomials and q-Genocchinumbers and polynomials. To be more precise we summarize our results asfollows, we obtain some combinatorial relations between q-Genocchi numbers andpolynomials with weight alpha and beta. Furthermore, we derive an integralrepresentation of weighted q-Bernstein polynomials of degree n on Zp. Also wededuce a fermionic p-adic q-integral representation of product weightedq-Bernstein polynomials of different degrees n1,n2,...on Zp and show that itcan be written with q-Genocchi numbers with weight alpha and beta which yieldsa deeper insight into the effectiveness of this type of generalizations. Ournew generating function possess a number of interesting properties which westate in this paper.

advances in difference equations

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