Krall-Jacobi commutative algebras of partial differential operators

Journal des Mathematiques Pures et Appliquees

Volumn 96, Issue 5, 2011, Pages 446-461

  Iliev, Plamen ^a,b  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b MAX PLANCK INSTITUT FÜR MATHEMATIK   (Germany)

Author keywords

Commutative algebras of partial differential operators; Darboux transformation; Krall's problem; Polynomials in several variables

Indexed keywords

EID: 80053624509     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2011.03.001     Document Type: Article

References (19)

1. Darboux G. Leçons sur la théorie générale des surfaces 1912, Gauthier-Villars.
2. Duistermaat J.J., Grünbaum F.A. Differential equations in the spectral parameter. Comm. Math. Phys. 1986, 103(2):177-240.
3. Dunkl C.F., Xu Y. Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications 2001, vol. 81. Cambridge University Press, Cambridge.
4. Grünbaum F.A., Haine L., Horozov E. Some functions that generalize the Krall-Laguerre polynomials. J. Comput. Appl. Math. 1999, 106(2):271-297.
5. Grünbaum F.A., Yakimov M. Discrete bispectral Darboux transformations from Jacobi operators. Pacific J. Math. 2002, 204:395-431.
6. Haine L. The Bochner-Krall problem: some new perspectives. NATO Sci. Ser. II Math. Phys. Chem. 2001, vol. 30:141-178. Kluwer Acad. Publ., Dordrecht.
7. Haine L., Iliev P. Commutative rings of difference operators and an adelic flag manifold. Internat. Math. Res. Notices 2000, 2000(6):281-323.
8. P. Iliev, Discrete versions of the Kadomtsev-Petviashvili hierarchy and the bispectral problem, Ph.D. thesis, Université Catholique de Louvain, 1999.
9. Iliev P. q-KP hierarchy, bispectrality and Calogero-Moser systems. J. Geom. Phys. 2000, 35(2-3):157-182.
10. Jacobi C.G.J. Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. J. Reine Angew. Math. 1859, 56:149-165.
11. Koekoek J., Koekoek R. Differential equations for generalized Jacobi polynomials. J. Comput. Appl. Math. 2000, 126(1-2):1-31.
12. Koornwinder T.H. Orthogonal polynomials with weight function (1-x)α(1+x)β+Mγ(x+1)+Nγ(x-1). Canad. Math. Bull. 1984, 27(2):205-214.
13. Krall H.L. Certain differential equations for the Tchebycheff polynomials. Duke Math. J. 1938, 4:705-718.
14. Krall H.L. On Orthogonal Polynomials Satisfying a Certain Fourth Order Differential Equation. The Pennsylvania Sate College Studies 1940, vol. 6.
15. Littlejohn L.L. The Krall polynomials: a new class of orthogonal polynomials. Quaestiones Math. 1982/1983, 5(3):255-265.
16. Matveev V.B., Salle M.A. Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics 1991, Springer-Verlag, Berlin.
17. Reach M. Generating difference equations with the Darboux transformation. Comm. Math. Phys. 1988, 119(3):385-402.
18. Xu Y. Sobolev orthogonal polynomials defined via gradient on the unit ball. J. Approx. Theory 2008, 152(1):52-65.
19. Zhedanov A. A method of constructing Krall's polynomials. J. Comput. Appl. Math. 1999, 107:1-20.