The generalized k-moment method is formulated in terms of Cutteridge-Devyatov polynomials (CDP). In this novel approach, the moments involved are spectral averages of integer powers of the logarithm of the absorption coefficient. The technique to obtain k-distributions from those generalized moments is detailed both theoretically and from a practical point of view. Its outputs are afterward assessed against reference data in several test cases of increasing complexity. Indeed, the first ones involve single lines in the Lorentz, Doppler and Voigt regimes. The most sophisticated situations investigated in this work concern applications of the method to high resolution LBL data for pure CO2 at temperatures between 300 K and 2300 K and at atmospheric pressure. In any case, the CDP solution to the generalized k-moment problem is found to provide very accurate results. The present technique outperforms our previous approach to k-moment model-ing of the cumulative distribution of absorption coefficients of gases that were based on first, second, first inverse and logarithmic moments, in all the situations investigated. Equations required to apply the model are provided in the paper, both over narrow bands and the full spectrum.