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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.

##### MSC:
 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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##### References:
 [1] Appell, P., Kampé de Fériet, J.: Fonctions Hypergéométriqes et Hypersphériques–Polynomes d’Hermite. Gauthier-Villars, Paris (1926) · JFM 52.0361.13 [2] Barrio, R., Peña, J.M., Sauer, T.: Three term recurrence for the evaluation of multivariate orthogonal polynomials. J. Approx. Theory 162, 407–420 (2010) · Zbl 1190.33011 [3] Berens, H., Schmid, H.J.: On the number of nodes of odd degree cubature formulae for integrals with Jacobi weights on a simplex. In: Espelid, T.O., Genz, A. (eds.) Numerical integration. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 357, pp. 37–44. Kluwer Acad., Dordrecht (1992) · Zbl 0751.41024 [4] de Boor, C.: B-form basics. In: Farin, G.E. (ed.) Geometric Modeling: Algorithms and New Trends, pp. 131–148. SIAM, Philadelphia (1987) [5] Bos, L.P.: Bounding the Lebesgue function for Lagrange interpolation in a simplex. J. Approx. Theory 38, 43–59 (1983) · Zbl 0546.41003 [6] Cooper, S., Waldron, S.: The diagonalisation of the multivariate Bernstein operator. J. Approx. Theory 117, 103–131 (2002) · Zbl 1020.41008 [7] Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6, 345–390 (1991) · Zbl 0742.76059 [8] Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge (2001) · Zbl 0964.33001 [9] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Higher Transcendental Functions. McGraw-Hill, New York (1953–1955) (three volumes) · Zbl 0052.29502 [10] Ismail, M., Masson, D.R.: Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mt. J. Math. 21, 359–375 (1991) · Zbl 0744.33004 [11] Kowalski, M.A.: The recursion formulas for orthogonal polynomials in n variables. SIAM J. Math. Anal. 13, 309–315 (1982) · Zbl 0494.33011 [12] Kowalski, M.A.: Orthogonality and recursion formulas for polynomials in n variables. SIAM J. Math. Anal. 13, 316–323 (1982) · Zbl 0497.33011 [13] Proriol, J.: Sur une famille de polynomes à variables orthogonaux dans un triangle. C. R. Acad. Sci. Paris 245, 2459–2461 (1957) · Zbl 0080.05204 [14] Suetin, P.K.: Orthogonal Polynomials in Two Variables. Analytical Methods and Special Functions, vol. 3. Gordon and Breach, New York (1999) (translation from Russian) · Zbl 1058.33501 [15] Waldron, S.: On the Bernstein–Bézier form of Jacobi polynomials on a simplex. J. Approx. Theory 140, 86–99 (2006) · Zbl 1098.33008 [16] Xu, Y.: On multivariate orthogonal polynomials. SIAM J. Math. Anal. 24, 783–784 (1993) · Zbl 0770.42016 [17] Xu, Y.: Monomial orthogonal polynomials of several variables. J. Approx. Theory 133, 1–37 (2005) · Zbl 1081.33020 [18] Xu, Y.: Fourier series and approximation on hexagonal and triangular domains. Constr. Approx. 31, 115–138 (2010) · Zbl 1181.42008
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