Efficient exact quadrature of regular solid harmonics times polynomials over simplices in R3

A generalization of a recently introduced recursive numerical method (Gumerov et al., 2023) for the exact evaluation of integrals of regular solid harmonics and their normal derivatives over simplex elements in $R^3$ is presented. The original Quadrature to Expansion (Q2X) method (Gumerov et al., 2023) achieves optimal per-element asymptotic complexity for computing $O\left(p_s^2\right)$ integrals of all regular solid harmonics bases with truncation degree $p_s$ by exploiting recurrence relations of the regular solid harmonics as well as the flatness and straightness of the faces and edges, respectively, of simplex elements. 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 per-element complexity for evaluating all regular solid harmonics and monomial density functions. The method is derived for 1- and 2- simplex elements in $R^3$ and can be used for the boundary element method and vortex methods coupled with the fast multipole method.