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Three term relations for a class of bivariate orthogonal polynomials. (English) Zbl 1367.33014

Summary: We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the explicit expression for the matrix coefficients in these three term relations are deduced. These matrices are diagonal or tridiagonal with entries computable from the one variable coefficients in the respective three term recurrence relation. Moreover, some interesting particular cases are considered.

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
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[1] Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions, 9th printing. Dover, New York (1972) · Zbl 0543.33001
[2] Agahanov, S.A.: A method of constructing orthogonal polynomials of two variables for a certain class of weight functions (Russian). Vestnik Leningrad Univ. 20, 5-10 (1965)
[3] Chihara, T.S.: An introduction to orthogonal polynomials, Mathematics and its Applications, vol. 13. Gordon and Breach, New York (1978) · Zbl 0389.33008
[4] Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables, 2nd edition, Encyclopedia of Mathematics and its Applications, vol. 155. Cambridge Univ. Press, Cambridge (2014) · Zbl 1317.33001 · doi:10.1017/CBO9781107786134
[5] Fernández, L., Pérez, T.E., Piñar, M.A.: On Koornwinder classical orthogonal polynomials in two variables. J. Comput. Appl. Math. 236, 3817-3826 (2012) · Zbl 1262.42007 · doi:10.1016/j.cam.2011.08.017
[6] Gautschi, W.: An algorithmic implementation of the generalized Christoffel Theorem. In: Hammerlin, G. (ed.) Numerical Integration, International Series of Numerical Mathematics, vol. 57, pp. 89-106. Birkhäuser, Basel (1982) · Zbl 0518.65006
[7] Koekoek, R., Lesky, P.A., Swarttouw, R.E.: Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics. Springer, Berlin (2010) · Zbl 1200.33012 · doi:10.1007/978-3-642-05014-5
[8] Karlin, S., McGregor, J.: The Hahn polynomials, formulas and an application. Scripta Mathematica 26, 33-46 (1961) · Zbl 0104.29103
[9] Koornwinder, T.H.: Two-variable analogues of the classical orthogonal polynomials. In: Askey, R.A. (ed.) Theory and Application of Special Functions. Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, The University of Wisconsin-Madison, pp. 435-495. Academic Press, New York (1975) · Zbl 0326.33002
[10] Krall, H.L., Frink, O.: A new class of orthogonal polynomials: the Bessel polynomials. Trans. Am. Math. Soc. 65, 100-115 (1949) · Zbl 0031.29701 · doi:10.1090/S0002-9947-1949-0028473-1
[11] Krall, H.L., Sheffer, I.M.: Orthogonal polynomials in two variables. Ann. Mat. Pura Appl. (4) 76, 325-376 (1967) · Zbl 0186.38602
[12] Kwon, K.H., Lee, J.K., Littlejohn, L.L.: Orthogonal polynomial eigenfunctions of second-order partial differential equations. Trans. Am. Math. Soc. 353, 3629-3647 (2001) · Zbl 0972.33007 · doi:10.1090/S0002-9947-01-02784-2
[13] Nikiforov, A.F., Suslov, S.K., Uvarov, V.B.: Classical Orthogonal Polynomials of a Discrete Variable, Springer series in Computational Physics. Springer, Berlin (1991) · Zbl 0743.33001 · doi:10.1007/978-3-642-74748-9
[14] Rodal, J., Area, I., Godoy, E.: Orthogonal polynomials of two discrete variables on the simplex. Integral Transforms Spec. Funct. 16, 263-280 (2005) · Zbl 1065.33013 · doi:10.1080/1065246042000272036
[15] Rodal, J., Area, I., Godoy, E.: Structure relations for monic orthogonal polynomials in two discrete variables. J. Math. Anal. Appl. 340, 825-844 (2008) · Zbl 1142.33004 · doi:10.1016/j.jmaa.2007.09.003
[16] Szegő, G.: Orthogonal polynomials, 4th edn. Amer. Math. Soc. Colloq. Publ., vol. 23. Amer Math Soc. Providence (1978) · JFM 65.0278.03
[17] Tratnik, M.V.: Multivariable Meixner, Krawtchouk, and Meixner-Pollaczek polynomials. J. Math. Phys. 30, 2740-2749 (1989) · Zbl 0696.33010 · doi:10.1063/1.528507
[18] Tratnik, M.V.: Some multivariable orthogonal polynomials of the Askey tableau-discrete families. J. Math. Phys. 32, 2337-2342 (1991) · Zbl 0742.33007 · doi:10.1063/1.529158
[19] Xu, Y.: A class of bivariate orthogonal polynomials and cubature formula. Numer. Math. 69, 233-241 (1994) · Zbl 0820.41023 · doi:10.1007/s002110050089
[20] Zhedanov, A.: Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math. 85(1), 67-86 (1997) · Zbl 0918.42016 · doi:10.1016/S0377-0427(97)00130-1
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