Fourth order partial differential equations for Krall-type orthogonal polynomials on the triangle
HTML articles powered by AMS MathViewer
- by Antonia M. Delgado, Lidia Fernández and Teresa E. Pérez PDF
- Proc. Amer. Math. Soc. 146 (2018), 3961-3974 Request permission
Abstract:
We construct bivariate polynomials orthogonal with respect to a Krall-type inner product on the triangle defined by adding Krall terms over the border and the vertexes to the classical inner product. We prove that these Krall-type orthogonal polynomials satisfy fourth order partial differential equations with polynomial coefficients, as an extension of the classical theory introduced by H. L. Krall in the 1940s.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
- R. Álvarez-Nodarse, J. Arvesú, and F. Marcellán, Modifications of quasi-definite linear functionals via addition of delta and derivatives of delta Dirac functions, Indag. Math. (N.S.) 15 (2004), no. 1, 1–20. MR 2061464, DOI 10.1016/S0019-3577(04)90001-8
- Antonia M. Delgado, Lidia Fernández, Teresa E. Pérez, Miguel A. Piñar, and Yuan Xu, Orthogonal polynomials in several variables for measures with mass points, Numer. Algorithms 55 (2010), no. 2-3, 245–264. MR 2720631, DOI 10.1007/s11075-010-9391-z
- Antonia M. Delgado, Lidia Fernández, Teresa E. Pérez, and Miguel A. Piñar, On the Uvarov modification of two variable orthogonal polynomials on the disk, Complex Anal. Oper. Theory 6 (2012), no. 3, 665–676. MR 2944078, DOI 10.1007/s11785-011-0192-8
- Antonia M. Delgado, Lidia Fernández, Teresa E. Pérez, and Miguel A. Piñar, Multivariate orthogonal polynomials and modified moment functionals, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 090, 25. MR 3545477, DOI 10.3842/SIGMA.2016.090
- Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 155, Cambridge University Press, Cambridge, 2014. MR 3289583, DOI 10.1017/CBO9781107786134
- Lidia Fernández, Teresa E. Pérez, Miguel A. Piñar, and Yuan Xu, Krall-type orthogonal polynomials in several variables, J. Comput. Appl. Math. 233 (2010), no. 6, 1519–1524. MR 2559340, DOI 10.1016/j.cam.2009.02.067
- Plamen Iliev, Krall-Jacobi commutative algebras of partial differential operators, J. Math. Pures Appl. (9) 96 (2011), no. 5, 446–461 (English, with English and French summaries). MR 2843221, DOI 10.1016/j.matpur.2011.03.001
- J. Koekoek and R. Koekoek, Differential equations for generalized Jacobi polynomials, J. Comput. Appl. Math. 126 (2000), no. 1-2, 1–31. MR 1806105, DOI 10.1016/S0377-0427(99)00338-6
- Tom Koornwinder, Two-variable analogues of the classical orthogonal polynomials, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 435–495. MR 0402146
- Tom H. Koornwinder, Orthogonal polynomials with weight function $(1-x)^{\alpha }(1+x)^{\beta }+M\delta (x+1)+N\delta (x-1)$, Canad. Math. Bull. 27 (1984), no. 2, 205–214. MR 740416, DOI 10.4153/CMB-1984-030-7
- H. L. Krall, On orthogonal polynomials satisfying a certain fourth order differential equation, Pennsylvania State College Studies 1940 (1940), no. 6, 24. MR 2679
- Allan M. Krall, Orthogonal polynomials satisfying fourth order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 87 (1980/81), no. 3-4, 271–288. MR 606336, DOI 10.1017/S0308210500015213
- H. L. Krall and I. M. Sheffer, Orthogonal polynomials in two variables, Ann. Mat. Pura Appl. (4) 76 (1967), 325–376. MR 228920, DOI 10.1007/BF02412238
- Lance L. Littlejohn, The Krall polynomials: a new class of orthogonal polynomials, Quaestiones Math. 5 (1982/83), no. 3, 255–265. MR 690030
- Clotilde Martínez and Miguel A. Piñar, Orthogonal polynomials on the unit ball and fourth-order partial differential equations, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 020, 11. MR 3463057, DOI 10.3842/SIGMA.2016.020
- Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1959. Revised ed. MR 0106295
- V. B. Uvarov, The connection between systems of polynomials that are orthogonal with respect to different distribution functions, Ž. Vyčisl. Mat i Mat. Fiz. 9 (1969), 1253–1262 (Russian). MR 262764
- Alexei Zhedanov, A method of constructing Krall’s polynomials, J. Comput. Appl. Math. 107 (1999), no. 1, 1–20. MR 1698475, DOI 10.1016/S0377-0427(99)00070-9
Additional Information
- Antonia M. Delgado
- Affiliation: Departamento de Matemática Aplicada & Instituto de Matemáticas (IEMath - GR) Universidad de Granada. 18071. Granada, Spain
- Email: amdelgado@ugr.es
- Lidia Fernández
- Affiliation: Departamento de Matemática Aplicada & Instituto de Matemáticas (IEMath - GR) Universidad de Granada. 18071. Granada, Spain
- Email: lidiafr@ugr.es
- Teresa E. Pérez
- Affiliation: Departamento de Matemática Aplicada & Instituto de Matemáticas (IEMath - GR) Universidad de Granada. 18071. Granada, Spain
- MR Author ID: 321333
- Email: tperez@ugr.es
- Received by editor(s): March 24, 2017
- Received by editor(s) in revised form: September 19, 2017, December 5, 2017, and December 12, 2017
- Published electronically: May 24, 2018
- Additional Notes: This work has been partially supported by MINECO of Spain and the European Regional Development Fund (ERDF) through grant MTM2014-53171-P, and by Junta de Andalucía grant P11-FQM-7276 and research group FQM-384.
The third author is the corresponding author. - Communicated by: Yuan Xu
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3961-3974
- MSC (2010): Primary 33C50, 42C05
- DOI: https://doi.org/10.1090/proc/14052
- MathSciNet review: 3825849