Summary: Asymptotic properties of the half-linear difference equation \[ \Delta (a_{n}| \Delta x_{n}| ^{\alpha }\text{sgn}\, \Delta x_{n} )=b_{n}| x_{n+1}| ^{\alpha }\text{sgn}\, x_{n+1} \tag{\(*\)} \] 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.