Recurrence relations for orthogonal polynomials on triangular domains

In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P_n,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.