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Differential operators having Sobolev type Laguerre polynomials as eigenfunctions


Author: H. Bavinck
Journal: Proc. Amer. Math. Soc. 125 (1997), 3561-3567
MSC (1991): Primary 33C45, 34A35
DOI: https://doi.org/10.1090/S0002-9939-97-04091-4
MathSciNet review: 1422848
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Abstract: We consider the polynomials $\left\{ L_n^{\alpha ,M}(x,l)\right\} _{n=0}^\infty $ orthogonal with respect to the Sobolev type inner product

\begin{equation*}\left\langle p,q\right\rangle =\frac 1{\Gamma (\alpha +1)}\int _0^\infty p(x)q(x)x^\alpha e^{-x}dx+Mp^{(l)}(0)q^{(l)}(0), \end{equation*}

where $\alpha >-1,M\geq 0$ and $l$ is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order $2\alpha +4l+4$ if $\alpha $ is a nonnegative integer and $M>0.$


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  • 1. H. Bavinck: A direct approach to Koekoek's differential equation for generalized Laguerre polynomials. Acta Math. Hungar. 66 (3) (1995), 247-253.MR 96a:33009
  • 2. H. Bavinck: Linear perturbations of differential or difference operators with polynomials as eigenfunctions. J. Comp. Appl. Math. 78 (1997), 179-195.
  • 3. H. Bavinck: A new result for Laguerre polynomials. J. Physics A. 29 (1996), L277-L279. CMP 96:15
  • 4. I.H. Jung, K.H. Kwon and G.J. Yoon: Differential equations of infinite order for Sobolev-type orthogonal polynomials. J. Comp. Appl. Math. 78 (1997), 277-293.
  • 5. A.M. Krall: Orthogonal polynomials satisfying fourth order differential equations. Proc. Roy. Soc. Edinburgh, 87A (1981), 271-288.MR 82d:33021
  • 6. A.M. Krall and L.L. Littlejohn: On the classification of differential equations having orthogonal polynomial solutions II. Ann. Mat. Pura Appl. (4) 149 (1987), 77-102. MR 89e:34017
  • 7. H.L. Krall: On orthogonal polynomials satisfying a certain fourth order differential equation. The Pennsylvania State College Studies, No. 6, The Pennsylvania State College, State College, Pa., 1940.
  • 8. L.L. Littlejohn: An application of a new theorem on orthogonal polynomials and differential equations. Quaestiones Math. 10 (1986), 49-61.MR 87m:42021
  • 9. J. Koekoek and R. Koekoek: On a differential equation for Koornwinder's generalized Laguerre polynomials. Proc. Amer. Math. Soc. 112, 1991, 1045-1054.MR 91j:33008
  • 10. J. Koekoek , R. Koekoek and H. Bavinck: On differential equations for Sobolev-type Laguerre polynomials. TUDelft Fac. Techn. Math. & Inf. Report 95-79. To appear in Trans. Amer. Math. Soc.
  • 11. R. Koekoek: Generalizations of the classical Laguerre polynomials. J. Math. Anal. Appl. 153 (1990), 576-590. MR 92d:33016
  • 12. R. Koekoek: Generalizations of the classical Laguerre polynomials and some $q$-analogues. Thesis Delft University of Technology 1990.
  • 13. T.H. Koornwinder: Orthogonal polynomials with weight function $\left( 1-x\right) ^\alpha \left( 1+x\right) ^\beta\break +M\delta (x+1)+N\delta (x-1).$ Canad. Math. Bull. 27(2), 1984, 205-214.MR 85i:33011
  • 14. F. Marcellán and A. Ronveaux: On a class of polynomials orthogonal with respect to a Sobolev inner product. Indag. Math. N.S. 1 (1990), 451-464.MR 92f:42029

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Additional Information

H. Bavinck
Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, Mekelweg 4, 2628 CD Delft, The Netherlands
Email: bavinck@twi.tudelft.nl

DOI: https://doi.org/10.1090/S0002-9939-97-04091-4
Keywords: Differential operators, Sobolev type Laguerre polynomials
Received by editor(s): June 27, 1996
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1997 American Mathematical Society

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