Single-cluster three-dimensional Laplace multipole evaluator

Fix an integer $p\ge 0$, a finite set of source labels $J$ with positions $y_j\in {\mathbb{R}}^3$ and charges $q_j\in \mathbb{C}$, a center $c$, and a radius $a\ge 0$ with $\mathrm{\mid }{y}_j-c\mathrm{\mid }\le a$. For each $0\le \mathrm{\ell }\le p$, supply an orthonormal basis $\{Y_{\mathrm{\ell }m}:-\mathrm{\ell }\le m\le \mathrm{\ell }\}$ of ${\mathcal{H}}_{\mathrm{\ell }}$ from Degree-indexed spherical harmonic spaces in three dimensions; its existence and dimension are given by Three-dimensional Laplace spherical-harmonic addition theorem. Form moments $M_{\mathrm{\ell }m}={\sum }_{j\in J}{q}_j\overline{H_{\mathrm{\ell }m}(y_j-c)}\mathrm{.}$ For every target $x$ with $r=\mathrm{\mid }x-c\mathrm{\mid }>a$, evaluate the truncated single-cluster multipole field ${\tilde{U}}_p(x)=4\pi {\sum }_{\mathrm{\ell }=0}^p\frac{1}{2\mathrm{\ell }+1}\frac{1}{r^{\mathrm{\ell }+1}}{\sum }_{m=-\mathrm{\ell }}^{\mathrm{\ell }}{M}_{\mathrm{\ell }m}{Y}_{\mathrm{\ell }m}\text{\hspace{0.17em}⁣}\left(\frac{x-c}{r}\right)\mathrm{.}$ The output is an approximation to $U(x)={\sum }_{j\in J}{q}_j\mathrm{/}\mathrm{\mid }x-y_j\mathrm{\mid }$. This is the source-cluster-to-target evaluator only; it makes no 3D tree-translation or global all-target complexity claim.

For the arithmetic count, store each basis polynomial by its monomial coefficient array. Basis setup is done once for the chosen $p$. To evaluate the displayed formula directly, evaluate every basis polynomial by summing its monomials at each source and target point, then accumulate the moments and target sums. This convention makes the basis-evaluation work explicit rather than treating each special-function evaluation as free.