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# Recurrence formulae for Rn,m and Sn,m

The associated Legendre function Pnm has the following recurrence formulae:

 (C.1) (C.2)

Multiplying (C.1) by one obtains (n+m+1)(n+1-m)Rn+1,m - (2n+1) x3 Rn,m + r2 Rn-1,m=0.

Multiplying (C.2) by , setting m=n+1 and using (A.2) one obtains

In the same manner, one has r2 Sn+1,m-(2n+1) x3 Sn,m + (n+m)(n-m)Sn-1,m=0 .

Using above formulae one can compute Rn,m and Sn,m as follows:
• Computation of Rn,m
1.
R0,0=1.
2.
Compute Rn,n recursively for n=1,2,....,P using
 (C.3)

3.
For a fixed m use the following formula:
 (n+m+1)(n+1-m)Rn+1,m - (2n+1) x3 Rn,m + r2 Rn-1,m=0 (C.4)

to compute Rn,m recursively for .
• Computation of Sn,m
1.
S0,0=1/r.
2.
Compute Sn,n recursively for n=1,2,....,P using

3.
For a fixed m use the following formula: r2 Sn+1,m-(2n+1) x3 Sn,m + (n+m)(n-m)Sn-1,m=0

to compute Sn,m recursively for .
• Functions Rn,m and Sn,m for a negative m can be computed via (A.5) and (A.6).

Next: Derivatives of and Sn,m Up: Applications of Fast Multipole Previous: Relations between solid harmonics Rn,m Sn,m
Ken-ichi Yoshida
2001-07-28