Asymptotic properties of the half-linear difference equation (∗) ∆(an|∆xn| α sgn ∆xn) = bn|xn+1| α sgn xn+1 are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to (∗) are considered too. Our approach is based on a classification of solutions of (∗) and on some summation inequalities for double series, which can be used also in other different contexts.