Recursive three term recurrence relations for the Jacobi polynomials on a triangle

Shayne Waldron


Given a suitable weight on $\R^d$, there exist many (recursive) three term recurrence relations for the corresponding multivariate orthogonal polynomials. In principle, these can be obtained by calculating pseudoinverses of a sequence of matrices. Here we give an explicit recursive three term recurrence for the multivariate Jacobi polynomials on a simplex. This formula was obtained by seeking the best possible three term recurrence. It defines corresponding linear maps, which have the same symmetries as the spaces of Jacobi polynomials on which they are defined. The key idea behind this formula is that some Jacobi polynomials on a simplex can be viewed as univariate Jacobi polynomials, and for these the recurrence reduces to the univariate three term recurrence.

Keywords: (recursive) three term recurrence relations, multivariate Jacobi polynomials, Legendre polynomials on a triangle, barycentric coordinates, symmetry group,

Math Review Classification: Primary 33C50, 33F05, 42C05; Secondary 05C90, 52B15

Length: 19 pages

Last Updated: 27 August 2009