Asymptotic form of the matrix elements of the direct collision integral in the Boltzmann equation

The asymptotic form is investigated for large indices of the matrix elements of the collision integral in the Boltzmann equation, which are the expansion coefficients of the collision integral in spherical Hermitian polynomials in the isotropic case for power interaction potentials. It is established that the principal term of the asymptotic expansion can be presented as the product of the power function of one of the indices by the homogeneous function of the ratios of the indices with the degree of homogeneity determined by the exponent in the interaction potential. The principal term of asymptotic expansion, which is the same for all ratios of the matrix element indices, is obtained for the matrix elements of the integral of direct collisions.