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The present paper deals with Bernstein polynomials and Frobenius-Eulernumbers and polynomials. We apply the method of generating function andfermionic p-adic integral representation on Zp, which are exploited to derivefurther classes of Bernstein polynomials and Frobenius-Euler numbers andpolynomials. To be more precise we summarize our results as follows, we obtainsome combinatorial relations between Frobenius-Euler numbers and polynomials.Furthermore, we derive an integral representation of Bernstein polynomials ofdegree n on Zp . Also we deduce a fermionic p-adic integral representation ofproduct Bernstein polynomials of different degrees n1, n2,...on Zp and showthat it can be written with Frobenius-Euler numbers which yields a deeperinsight into the effectiveness of this type of generalizations. Ourapplications possess a number of interesting properties which we state in thispaper.

A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials