Recurrence relations for a family of orthogonal polynomials on a triangle

Abstract

This paper derives sparse recurrence relations between orthogonal polynomials on a triangle and their partial derivatives, which are analogous to recurrence relations for Jacobi polynomials. We derive these recurrences in a systematic fashion by introducing ladder operators that map an orthogonal polynomial to another by incrementing or decrementing its associated parameters by one. We apply the results to efficiently calculating the Laplacian of polynomial approximations of functions on the triangle, using polynomial degrees in the thousands, i.e., millions of degrees of freedom.