The exact solution of the auroral ion velocity distribution for the Bhatnagar Gross and Krook (B.G.K.) collision model is revised in order to obtain insights on convergence and approximation of the generalized orthogonal polynomial solution of the Boltzmann equation. Starting from a Maxwellian weighting function we show that the accuracy of the expansion is very sensitive to the assigned 'temperature'. This leads us to propose a generalized Grad's expansion solution of the Boltzmann equation that, without increasing either the number of calculated coefficients or the complexity of their equations, significantly enhances the accuracy of the limited expansion. This method when applied to the relaxation model increases the range of electric fields and enlarges the region of the velocity space where the limited expansion is valid.