Asymptotic properties of solutions of difference equation of the form Ξm(xnβ+unβxn+kβ)=anβf(n,xΟ(n)β)+bnβ are 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