Semi-classical expansion of distribution function using modified Hermite polynomials for quantum gas

The author proposes the semi-classical expansion of the distribution function using modified Hermite polynomials to calculate moment equations for quantum gas. The completeness of the semi-classical expansion of the distribution function is not satisfied, whereas we can conjecture that moment equations obtained using the semi-classical expansion coincides with those obtained using Uehling-Uhlenbeck equation. Actually, Grad's 13 moment equations, which are calculated using correct Grad's 13 moment equation, coincide with those, which are calculated using the semi-classical expansion of the distribution function, when the collisional term of the Uehling-Uhlenbeck equation is replaced with the quantum Bhatnagar-Gross-Krook model.