Recurrence Relations for Orthogonal Polynomials on Triangular Domains

In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P-n,P-r(u, v, w), r = 0, 1,..., n, n >= 0 on the triangular domain T = {(u, v, w) : u, v, w >= 0, u + v + w = 1} are constructed, where u, v, w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation.