Efficient Exact Quadrature of Regular Solid Harmonics Times Polynomials Over Simplices in $\mathbb{R}^3$

A generalization of a recently introduced recursive numerical method for the exact evaluation of integrals of regular solid harmonics and their normal derivatives over simplex elements in $\mathbb{R}^3$ is presented. The original Quadrature to Expansion (Q2X) method achieves optimal per-element asymptotic complexity, however, it considered only constant density functions over the elements. Here, we generalize this method to support arbitrary degree polynomial density functions, which is achieved in an extended recursive framework while maintaining the optimality of the complexity. The method is derived for 1- and 2- simplex elements in $\mathbb{R}^3$ and can be used for the boundary element method and vortex methods coupled with the fast multipole method.