# Recurrence Relations for Orthogonal Polynomials on Triangular Domains

Department of Mathematics, Jordan University of Science and Technology, Irbid 22110, Jordan

Academic Editors: Zhongqiang Zhang and Mohsen Zayernouri

Received: 26 December 2015 / Revised: 15 March 2016 / Accepted: 7 April 2016 / Published: 12 April 2016

(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)

In Farouki et al , 2003, Legendre-weighted orthogonal polynomials ${\mathcal{P}}_{n,r}(u,v,w),r=0,1,\dots ,n,\phantom{\rule{0.166667em}{0ex}}n\ge 0$ on the triangular domain $T=\left\{\right(u,v,w):u,v,w\ge 0,u+v+w=1\}$ 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.