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.

翻译：本文提出了一种近期引入的递归数值方法的推广形式，用于精确计算$\mathbb{R}^3$中单纯形单元上正则固体调和函数及其法向导数的积分。原始的"求积到展开"（Q2X）方法实现了每个单元的最优渐近复杂度，但仅考虑了单元上的常数密度函数。在此，我们将该方法推广至支持任意次数的多项式密度函数，这一推广在扩展的递归框架中实现，同时保持了复杂度的最优性。该方法针对$\mathbb{R}^3$中的1-和2-单纯形单元推导，可用于结合快速多极子方法的边界元方法和涡方法。