Consider the standard multipole expansion for the coulomb potential often used in the fast multipole method algorithm (Theorem 5.2). A translation theorem is used in the algorithm (Theorem 5.3).
Is O(n,m) the same as M(n,m) in Theorem 5.2 except shifted with coordinates rho', alpha', and beta'? This makes sense but I wanted to double check.
When evaluating the expansion in Theorem 5.3, the factorial terms in A(j-n,k-m) will contain negative integers (namely from k-m), giving an undefined result or complex infinity. Additionally, the O(j-n,k-m) term will be impossible to evaluate due to the spherical harmonic indices being impossible (in Y(n,m), n must be greater or equal to abs(m) and this is not the case for indices (j-n,k-m)). What do we do about this? How is Theorem 5.3 evaluated correctly?
source for Theorems: https://math.nyu.edu/faculty/greengar/shortcourse_fmm.pdf