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Recursive three-term recurrence relations for the Jacobi polynomials on a triangle. (English) Zbl 1247.33023
Given a suitable weight on \(\mathbb{R}^d\), there exist many (recursive) three-term recurrence relations for the corresponding multivariate orthogonal polynomials. These can be obtained by calculating pseudoinverses of a sequence of matrices. The author gives an explicit recursive three-term recurrence relation for the multivariate Jacobi polynomials on a simplex. This formula is obtained by seeking the best possible three-term recurrence relation. This 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 relation reduces to the univariate three-term recurrence relation.

33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C65 Appell, Horn and Lauricella functions
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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