A fast multipole method for stellar dynamics

Table 1 The FMM kernels

Name	Meaning	Equation
P2P	particle to particle	(1)
P2M	particle to multipole	(3c)
M2M	multipole to multipole	(3d)
M2P	multipole to particle	(3g)
M2L	multipole to local expansion	(3b)
P2L	particle to local expansion	see table legend
L2L	local expansion to local expansion	(3e)
L2P	local expansion to particle	(3f)

The tree code replaces direct summation (P2P) with P2M-M2M-M2P, while FMM uses P2M-M2M-M2L-L2L-L2P, see also Figure 2. The P2L kernel corresponds to equation (3b) with $z_{A} \to x_{a}$ and $M_{k}^{l} \to m_{a} δ_{k 0}$ , i.e. $F_{n}^{m} (s_{B}) = m_{a} Θ_{n}^{m} (s_{B} - x_{a})$ .