×

A construction of real weight functions for certain orthogonal polynomials in two variables. (English) Zbl 1096.35029

Real orthogonalizing weight functions for the orthogonal polynomials satisfying to the following four types of partial differential equations
\[ \begin{gathered} x^2u_{xx}+2xyu_{xy}+(y^2-y)u_{yy}+g(x-1)u_x+g(y-\gamma)u_y=\lambda y\\ (x^2+y)u_{xx}+2xyu_{xy}+y^2u_{yy}+gxu_x+g(y-1)u_y=\lambda y\\ (x^2-x)u_{xx}+2xyu_{xy}+y^2u_{yy}+(dx+e)u_x+(dy+h)u_y=\lambda y\\ xu_{xx}+2yu_{xy}+(dx+e)u_x+(dy+h)u_y=\lambda y\\ \end{gathered} \] are constructed. The first and the second of these equations were considered earlier by H. L. Krall and I. M. Sheffer in [Ann. Mat. Pura Appl., IV. Ser. 76, 325–376 (1967; Zbl 0186.38602)].
In Example 5.1 it is proved that the first equation above has a bivariate polynomial system of the form \(\psi_{n- k,k}(x,y)= B_{n-k}^{(g+2k, -g)}(x)x^k L_k^{(g\gamma- 1)}(gy/x), \,k=0, \dotsc,n\), where \(B_{k}^{(g+2k,-g)}\) and \(L_k^{(g\gamma- 1)}\) are Bessel and Laguerre polynomials. The function \(\omega(x,y)=\bar\omega^{(g- 1,-g)}(x) e^{-gy/x}x^{1- g\gamma} y^{g\gamma-1}\), where \((t^2\bar\omega^{(a,b)}(t))'= (at+b)\bar\omega^{(a,b)}(t)+ G^{(a,b)}(t)\) is a polynomial killer and \(G^{(a,b)}(t)\) is a weight function for the orthogonal polynomial system \(\{\psi_{n- k,k}(x,y)\}\).

MSC:

35C05 Solutions to PDEs in closed form
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
35G05 Linear higher-order PDEs

Citations:

Zbl 0186.38602
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berg, C.; Christensen, J. P.R.; Jensen, C. U., A remark on the multidimensional moment problem, Math. Anal., 243, 163-169 (1979) · Zbl 0416.46003
[2] Bochner, S., Über Sturm-Liouvillesche Polynomsysteme, Math. Z., 29, 730-736 (1929) · JFM 55.0260.01
[3] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach: Gordon and Breach New York · Zbl 0389.33008
[4] Duran, A., The Stieltjes moment problem for rapidly decreasing functions, Proc. Amer. Math. Soc., 107, 731-741 (1989) · Zbl 0676.44007
[5] Duran, A., Functions with given moments and weight functions for orthogonal polynomials, Rocky Mountain J. Math., 23, 87-104 (1993) · Zbl 0777.44003
[6] Gel’fand, I. M.; Shilov, G. E., Generalized Functions, vol. 1 (1964), Academic Press: Academic Press New York · Zbl 0115.33101
[7] Kim, Y. J.; Kwon, K. H.; Lee, J. K., Orthogonal polynomials in two variables and second order partial differential equations, J. Comput. Appl. Math., 82, 239-260 (1997) · Zbl 0889.33007
[8] Krall, H. L.; Sheffer, I. M., Orthogonal polynomials in two variables, Ann. Mat. Pura Appl., 4, 325-376 (1967) · Zbl 0186.38602
[9] Kwon, K. H.; Kim, S. S.; Han, S. S., Orthogonality of Tchebychev sets of polynomials, Bull. London Math. Soc., 24, 361-367 (1992) · Zbl 0768.33007
[10] Kwon, K. H.; Lee, J. K.; Littlejohn, L. L., Orthogonal polynomial eigenfunctions of second order partial differential equations, Trans. Amer. Math. Soc., 353, 3629-3647 (2001) · Zbl 0972.33007
[11] Kwon, K. H.; Littlejohn, L. L., Classification of classical orthogonal polynomials, J. Korean Math. Soc., 34, 973-1008 (1997) · Zbl 0898.33002
[12] Lesky, P., Die Charakterisierung der klassischen orthogonalen Polynome durch Sturm-Liouvillesche Differentialgleichungen, Arch. Ration. Mech. Anal., 10, 341-352 (1962) · Zbl 0105.27603
[13] Littlejohn, L. L., On the classification of differential equations having orthogonal polynomial solutions, Ann. Mat. Pura Appl., 4, 35-53 (1984) · Zbl 0571.34003
[14] Littlejohn, L. L., Orthogonal polynomial solutions to ordinary and partial differential equations, (Lecture Notes in Math., vol. 1329 (1988)), 98-124 · Zbl 0653.42022
[15] Maroni, P., An integral representation for the Bessel form, J. Comput. Appl. Math., 57, 251-260 (1995) · Zbl 0827.33006
[16] Morton, R. D.; Krall, A. M., Distributional weight functions for orthogonal polynomials, SIAM J. Math. Anal., 9, 604-626 (1978) · Zbl 0389.33009
[17] Stieltjes, T. J., Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, 8, J1-122 (1894), 9 (1895) A1-47 · JFM 25.0326.01
[18] Widder, D. V., The Laplace Transform (1941), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0060.24801
[19] Xu, Y., A class of bivariate orthogonal polynomials and cubature formula, Numer. Math., 69, 231-241 (1994) · Zbl 0820.41023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.