Generalization of the Hecke theorem for the nonlinear Boltzmann collision integral in the axisymmetric case

The properties of the nonlinear collision integral in the Boltzmann equation are studied. Expansions in spherical Hermitean polynomials are used. It was shown [1] that the nonlinear matrix elements of the collision operator are related to each other by simple expressions, which are valid for arbitrary cross sections of particle interaction. The structure of the collision operator and the properties of the matrix elements are studied for the case when the interaction potential is spherically symmetric. In this case, the linear Boltzmann operator satisfies the Hecke theorem. The generalized Hecke theorem, from which it follows that many nonlinear matrix elements vanish, is proved with recurrence relations derived. It is shown that the generalized Hecke theorem is a consequence of the ordinary Hecke theorem.