This paper is to establish new results on the zeros of \(F^{(k)}-\alpha (z)\) , where F(z) is a differential polynomial or difference polynomial of f and \(\alpha (z)\) is a small function with respect to f in the sense of Nevanlinna theory. We also obtain that at least one of \(F^{(k)}-\alpha (z)\) and \(G^{(k)}-\alpha (z)\) has infinitely many zeros, where F(z) and G(z) are crossed differential polynomials or difference polynomials of f and g.
