An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

Journal of Mathematical Analysis and Applications

Volumn 359, Issue 1, 2009, Pages 352-367

  Gómez Ullate, David ^a   Kamran, Niky ^b   Milson, Robert ^c  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

^b MCGILL UNIVERSITY   (Canada)

^c DALHOUSIE UNIVERSITY   (Canada)

Author keywords

Generalized Bochner theorem; Orthogonal polynomials

Indexed keywords

EID: 67649871973     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2009.05.052     Document Type: Article

References (29)

1. Aczel J. Eine Bemerkung über die Charakterisierung der klassichen orthogonale Polynome. Acta Math. Acad. Sci. Hungar. 4 (1953) 315-321
2. Alfaro M., Álvarez de Morales M., and Rezola M.L. Orthogonality of the Jacobi polynomials with negative integer parameters. J. Comput. Appl. Math. 145 (2002) 379-386
3. Askey R.A., and Wilson J.A. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 319 (1985)
4. Atkinson F.V., and Everitt W.N. Orthogonal polynomials which satisfy second order differential equations. E.B. Christoffel. Aachen/Monschau, 1979 (1981), Birkhäuser, Basel 173-181
5. Bochner S. Über Strum-Liouvillsche Polynomsysteme. Math. Z. 29 (1929) 730-736
6. n for positive integers k. J. Comput. Appl. Math. 171 (2004) 199-234
7. 1-Jacobi orthogonal polynomials. arXiv:0812.0728 [math.CA]
8. 1-Laguerre orthogonal polynomials. arXiv:0812.3559 [math.CA]
9. Everitt W.N., Kwon K.H., Littlejohn L.L., and Wellman R. Orthogonal polynomial solutions of linear ordinary differential equations. J. Comput. Appl. Math. 133 (2001) 85-109
10. Feldmann J. On a characterization of classical orthogonal polynomials. Acta Sci. Math. 17 (1956) 129-133
11. Gómez-Ullate D., Kamran N., and Milson R. An extension of Bochner's problem: Exceptional invariant subspaces. arXiv:0805.3376 [math-ph]
12. Grünbaum A., and Haine L. The q-version of a theorem of Bochner. J. Comput. Appl. Math. 68 (1996) 103-114
13. Heine E. Theorie der Kugelfunctionen und der verwandten Functionen (1878), Reimer, Berlin
14. Hendriksen E., and van Rossum H. Semiclassical orthogonal polynomials. Orthogonal Polynomials and Applications. Bar-le-Duc, 1984. Lecture Notes in Math. vol. 1171 (1985), Springer, Berlin 354-361
15. Ismail M.E.H. A generalization of a theorem of Bochner. J. Comput. Appl. Math. 159 (2003) 319-324
16. Ismail M.E.H., and van Assche W. Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia Math. (2005), Cambridge Univ. Press, Cambridge
17. Krall H.L. On Orthogonal Polynomials Satisfying a Certain Fourth Order Differential Equation. Pennsylvania State College Stud. vol. 6 (1940)
18. Kwon K.H., and Littlejohn L.L. Classification of classical orthogonal polynomials. J. Korean Math. Soc. 34 (1997) 973-1008
19. Lesky P. Die Charakterisierung der klassischen orthogonalen Polynome durch SturmLiouvillesche Differentialgleichungen. Arch. Ration. Mech. Anal. 10 (1962) 341-352
20. Mikolás M. Common characterization of the Jacobi, Laguerre and Hermite-like polynomials. Mat. Lapok 7 (1956) 238-248 (in Hungarian)
21. Quesne C. Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry. J. Phys. A 41 39 (2008) 392001
22. Routh E.J. On some properties of certain solutions of a differential equation of the second order. Proc. London Math. Soc. 16 (1885) 245-261
23. Reed M., and Simon B. Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness (1975), Academic Press, New York
24. Ronveaux A. Sur l'équation différentielle du second ordre satisfaite par une classe de polynômes orthogonaux semi-classiques. C. R. Acad. Sci. Paris Sér. I Math. 305 5 (1987) 163-166
25. Ronveaux A. Polynômes orthogonaux dont les polynômes dérivés sont quasi orthogonaux. C. R. Acad. Sci. Paris Sér. A-B 289 7 (1979) A433-A436
26. Ronveaux A., and Marcellán F. Differential equation for classical-type orthogonal polynomials. Canad. Math. Bull. 32 4 (1989) 404-411
27. Stieltjes T.J. Sur certains polynômes qui vérifient une équation différentielle du second ordre et sur la théorie des fonctions de Lamé. Acta Math. 6 (1885) 321-326
28. Szegö G. Orthogonal Polynomials. Colloq. Publ. vol. 23 (1939), Amer. Math. Soc., Providence, RI
29. Uvarov V.B. The connection between systems of polynomials that are orthogonal with respect to different distribution functions. USSR Comp. Math. Math. Phys. 9 (1969) 25-36