In this paper we establish the asymptotical representations of PðlÞ solutions of k-th order Emden -Fowler difference equationi¼0 jD i y nþsi j si signðy nþs0 Þ; where p n is close to a power sequence. We obtained the necessary conditions for the existence of PðlÞ solutions and we also found the sufficient conditions of their existence for non-critical classes. These results are generalizations of corresponding theorems for second-order Emden-Fowler equation.

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