For 
, the arbitrary term from
the kernel of 
can be chosen so as to
satisfy the biorthonormality condition (
).
For 
, it is necessary to
show that the multipole functions generated by the the
recursion relations () are consistent with
the biorthonormality condition. This is done inductively from
the zeroth order biorthonormality condition.
Using the identity
we have for n>0,
Consider the first two terms of this sum:
Here we have used the convention established earlier that 
. The second two terms can be arranged in the following way:
by the inductive hypothesis, and so 
for all n>0. 