We will now present formulae for calculating 
, 
and 
. We do this by expanding (
) and
() in the 
and 
basis (zeroth order basis) with the following definition:
We now define an operator 
so that
where 
is the projection onto the orthogonal
complement of 
. In other words, if
the domain of 
is restricted to make 
one to one, then 
is the
inverse of 
.
By applying 
on equation (), the
multipole functions can be generated according to
Or expressed in the zeroth order basis for 
,
where
are the matrix elements of the operator 
, and
are the eigenvalues of 
.
These relations will generate the multipole functions from the
seeds 
. If
then 
and
may be chosen arbitrarily subject to
To complete the prescription, we need a formula for calculating
the 
, and we also need to
check that the biorthonormality condition ()
is satisfied in the case 
.
For convenience of notation, we will define
for every n less than zero. This means that all the power series expansions hold for every integer n.
From (),
Expanding this in powers of 
, one gets
The second term can be expressed as
The second two terms vanish because of the biorthonormality
condition () and the eigenvalue equation
(). The final step is to evaluate this
equation by the recursion equation ():
A similar calculation using the adjoint equation gives an
alternative formula for 
.
Thus the problem has been reduced to evaluating the matrix
element 
.
For example, if the spectrum of 
is non-degenerate,
the first few coefficients of the recursion series are
In this case, the arbitrary additive terms are proportional to
. The rank n tensors 
are the coefficients of these terms, and may be chosen to be zero.
