Relations between matrix elements of the nonlinear Boltzmann collision integral in the axisymmetric case

The properties of the nonlinear Boltzmann collision integral for the axisymmetric velocity distribution are studied. Expansions in spherical Hermitian polynomials orthogonal to the Maxwellian weighting function are employed. It is shown that the nonlinear matrix elements of the collision operator are related to each other by simple relations, which are valid for arbitrary cross sections of particle interaction even if a preferential direction exists. The relations are derived from the invariance of the collision operator under the choice of basis functions or, more precisely, under both temperature and the mean velocity of the Maxwellian weighting function. The recurrent relations found allow one to calculate the matrix elements at large values of indices. This makes it possible to construct exact solutions to complicated kinetic problems.