Irregular solid harmonic ($r^{-n-1}{Y}_{nm}=r^{-(2n+1)}{H}_n$)

An irregular solid harmonic is harmonic on punctured three-dimensional Euclidean space and generally singular at the origin. The Kelvin transform maps a degree-$n$ regular solid harmonic to a constant multiple of its irregular counterpart.

A degree-$n$ homogeneous polynomial satisfies Euler homogeneous function theorem $\mathbf{r}\cdot \nabla H_n=n{H}_n$. The product rule gives $\Delta (r^s{H}_n)=r^s\Delta H_n+s(2n+s+1)r^{s-2}{H}_n$ in three dimensions. Taking $s=-(2n+1)$ gives the displayed identity and the equivalence of harmonicity on $r>0$. The origin is excluded; even $1/r$ has a distributional Dirac delta function source there.