If the eigenvalue problem for 
is solved for a
particular form of the collision operator, then it would be
useful to have a scheme to compute the derivatives of 
, 
and 
around 
to compute the nonhydrodynamic
effects described by 
. The scheme one would
use for this is similar to the recursion scheme outlined in
Kumar et al. [1980],
that is the swarm theory analogue of
the Chapman-Enskog method.
The eigenfunctions and eigenvalues of equation
(
) are expanded as a power series in 
:
Upon substituting these series into equation (),
the following recursion relations for multipole functions 
, 
are generated:
The recursion relations () define 
up to an arbitrary additive term in the
kernel of 
, i.e. a term
proportional to 
. The corresponding
additive term for 
is then fixed by the
normalization (). It will be shown that,
whilst the functions 
and 
may be quite different functions with
different choices of the arbitrary additive terms, the product
is unique.
Since these functions enter the Greens function through this
product, this means that any physical results are well
determined by this method.
