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 ,
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.
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.