  The transport of charged particles or photons in a scattering medium can be
modelled with a Boltzmann equation. The mathematical treatment for scattering
in such scenarios is often simplified if evaluated in a frame where the
scattering centres are, on average, at rest. It is common therefore, to use a
mixed coordinate system, wherein space and time are measured in a fixed
inertial frame, while momenta are measured in a "co-moving" frame. To
facilitate analytic and numerical solutions, the momentum dependency of the
phase-space density may be expanded as a series of spherical harmonics,
typically truncated at low order. A method for deriving the system of equations
for the expansion coefficients of the spherical harmonics to arbitrary order is
presented in the limit of isotropic, small-angle scattering. The method of
derivation takes advantage of operators acting on the space of spherical
harmonics. The matrix representations of these operators are employed to
compute the system of equations. The computation of matrix representations is
detailed and subsequently simplified with the aid of rotations of the
coordinate system. The eigenvalues and eigenvectors of the matrix
representations are investigated to prepare the application of standard
numerical techniques, e.g. the finite volume method or the discontinuous
Galerkin method, to solve the system.
