Relations between solid harmonics

one can transform as follows:

where we have used (A.5) and the chain rule:

The underlined part (a) can be rewritten as follows:

In the case where

In the case where *m*' < *m*

As a result, the underlined part (a) can be rewritten as

Hence one obtains

Now take two points

From (B.5) we obtain the following formula:

Substituting (B.7) into the left-hand side of (B.6), one obtains

Setting

Exchanging the order of the summation in (B.9) ( , See Fig.B), one obtains

Finally, comparing the right-hand side of (B.6) with that of (B.10) one obtains the following formula: