Publication: An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
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2009-11-01
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Elsevier
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We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X(1)-Jacobi and X(1)-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [-1, 1] or the half-line [0, infinity), respectively, and they are a basis of the corresponding L(2) Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions {p(i)}(i=1)(infinity), then it must be either the X(1)-Jacobi or the X(1)-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the X(1) polynomial sequences.
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© 2009 Elsevier Inc. All rights reserved.
We are grateful to Jorge Arvesú, Mourad Ismail, Francisco Marcellán and André Ronveaux for their helpful comments. A special note of thanks goes to Norrie Everitt for his suggestions and remarks regarding operator domains and the limit point/circle analysis, and to Lance Littlejohn for comments regarding classical polynomials with negative integer parameters. The research of DGU is supported in part by the Ramón y Cajal program of the Spanish ministry of Science and Technology and by the DGI under grants MTM2006-00478 and MTM2006-14603. The research of NK is supported in part by NSERC grant RGPIN 105490-2004. The research of RM is supported in part by NSERC grant RGPIN-228057-2004.
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[1] J. Aczel, Eine Bemerkung über die Charakterisierung der klassichen orthogonale Polynome, Acta Math. Acad.Sci. Hungar 4 (1953), 315-321.
[2] M. Alfaro, M. Álvarez de Morales, M. L. Rezola, Orthogonality of the Jacobi polynomials with negative integer parameters, Journal of Computational and Applied Mathematics, 145 (2002) 379–386.
[3] R.A. Askey and J.A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs American Mathematical Society No. 319 (1985).
[4] F.V. Atkinson and W.N. Everitt, Orthogonal polynomials which satisfy second order differential equations. E. B. Christoffel (Aachen/Monschau, 1979), pp. 173–181, Birkhuser, Basel-Boston, Mass., 1981.
[5] S. Bochner, Über Strum-Liouvillsche Polynomsysteme, Math. Z. 29 (1929), 730-736.
[6] W. N. Everitt, L. L. Littlejohn, R. Wellman, The Sobolev orthogonality and spectral analysis of the Laguerre polynomials L −k n for positive integers k, J. Comput. Appl. Math. 171 (2004), 199–234.
[7] W. N. Everitt, Note on the X1-Jacobi orthogonal polynomials, arXiV CA 0812.0728 and Note on the X1-Laguerre orthogonal polynomials, arXiV CA 0812.3559
[8] W. N. Everitt, K. H. Kwon, L. L. Littlejohn and R. Wellman, Orthogonal polynomial solutions of linear ordinary differential equations, J. Comp. Appl. Math 133 (2001), 85109.
[9] J. Feldmann, On a characterization of classical orthogonal polynomials, Acta. Sc. Math. 17 (1956), 129133.
[10] D. Gómez-Ullate, N. Kamran, and R. Milson, An extension of Bochner’s problem: exceptional invariant subspaces arXiV math-ph 0805.3376
[11] A. Grünbaum and L. Haine, The q-version of a theorem of Bochner, J. Comput. Appl. Math. 68 (1996), 103–114.
[12] E. Heine, Theorie der Kugelfunctionen und der verwandten Functionen, Berlin, 1878.
[13] E. Hendriksen and H. van Rossum, Semiclassical orthogonal polynomials, in “Orthogonal polynomials and applications” (Bar-le-Duc, 1984), 354–361, Lecture Notes in Math., 1171, Springer, Berlin, 1985.
[14] M.E.H. Ismail , A generalization of a theorem of Bochner, J. Comp. Appl. Math. 159 (2003), 319–324.
[15] M.E.H. Ismail and W. van Assche, Classical and quantum orthogonal polynomials in one variable, Encyclopedia in Mathematics, Cambridge University Press, Cambridge, 2005.
[16] H. L. Krall, On orthogonal polynomials satisfying a certain fourth order differential equation, The Pennsylvania State College Studies, No 6, 1940.
[17] K. H. Kwon and L. L. Littlejohn, Classification of classical orthogonal polynomials, J. Korean Math. Soc. 34 (1997), 973–1008.
[18] P. Lesky, Die Charakterisierung der klassischen orthogonalen Polynome durch SturmLiouvillesche Differentialgleichungen, Arch. Rat. Mech. Anal. 10 (1962), 341–352.
[19] M. Mikolás, Common characterization of the Jacobi, Laguerre and Hermitelike polynomials(in Hungarian), Mate. Lapok 7 (1956), 238–248.
[20] C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A: Math. Theor. 41 No. 39, 392001.
[21] E.J. Routh, On some properties of certain solutions of a differential equation of the second order, Proc. London Math. Soc., 16 (1885), 245-261.
[22] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York-London, 1975.
[23] A. Ronveaux, 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 (1987), no. 5, 163–166.
[24] A. Ronveaux, Polynômes orthogonaux dont les polynômes dérivés sont quasi orthogonaux, C. R. Acad. Sci. Paris S´er. A-B 289 (1979), no. 7, A433–A436.
[25] A. Ronveaux and F. Marcell´an, Differential equation for classical-type orthogonal polynomials, Canad. Math. Bull. 32 (1989), no. 4, 404–411.
[26] T.J. Stieltjes, Sur certains polynômes qui vérifient une équation différentielle du second ordre et sur la théorie des fonctions de Lam´e, Acta Mathematica 6 (1885), 321-326.
[27] G. Szegö, Orthogonal polynomials, Colloquium Publications 23, American Mathematical Society, Providence, 1939.
[28] V. B. Uvarov, The connection between systems of polynomials that are orthogonal with respect to different distribution functions, USSR Computat. Math. and Math. Phys. 9 (1969), 25–36.