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Asymptotic properties of solutions of difference equation of the form \[\Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n \] are studied. We givesufficient conditions under which all solutions, or all solutions withpolynomial growth, or all nonoscillatory solutions are asymptoticallypolynomial. We use a new technique which allows us to control the degree ofapproximation.

Asymptotically polynomial solutions of difference equations of neutral type