Asymptotically polynomial solutions to difference equations of neutral type

Asymptotic properties of solutions to difference equation of the form Δm(xn+unxn+k)=anf(n,xσ(n))+bnΔm(xn+unxn+k)=anf(n,xσ(n))+bnare studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or all nonoscillatory solutions are asymptotically polynomial. We use a new technique which allows us to control the degree of approximation.